Math Easy Solutions

In #911Truth Part 7 I go over the strategy employed by Controlled Opposition Groups and their main goal: stray people away from looking at the facts as much as possible because the Truth cuts right through to your subconscious. To understand how and why controlled opposition groups work, it is first important to see just how powerful and ignored our Subconscious, or known as “Gut”, Intuition, or even just “BS Detector”, really is. When people are presented with information, often times they will have a distinct “feeling” towards that information, either positive or negative even if they are not consciously aware of it; much like driving a car in which you daze off but still get to your destination in one piece. Controlled Opposition groups work by mixing in real and false information so that the people that can see through the blatant mainstream lies are still less likely to jump on board the alternative mixture of lies, because the subconscious still knows something is not right. Thus other factors play a larger role such as “Authority”, Social Stigma, and if having an opinion on the topic is even worth it. But when someone is presented with the absolute unmuddied truth, the subconscious can feel right away to wake up your conscious brain and investigate further! This was exactly my experience when I heard about the simple question “Where Did The Towers Go?” in which my entire life flashed before my eyes and I felt the need to explore further into 9/11; something which wasn’t the case when I heard simple chants of “9/11 was an inside job”. And when #FreeEnergy was tied directly with 9/11, I knew from that moment on that there was no going back and my lifelong goal of developing #FreeEnergy has just begun! My experience is a good example of why Controlled Opposition groups are literally anywhere and everywhere!` Some examples of almost certain controlled psychological operation groups that I cover in this video is Alex Jones, InfoWars, “AlienScientist”, David Icke, “Flat Eart”, and the highly coordinated disinformation campaign regarding the California Forest Fires being used to mix in Directed Energy Weapons as a way to discredit from the irrefutable 9/11 – DEW connection. What makes these groups almost certainly controlled is not necessarily what information they are pushing but rather the information they leave out or purposely spread misinformation about, and just so happens to be about the most important event of our lifetimes: 9/11. Not only that but the site traffic of these websites are MUCH larger than actual truth website such as DrJudyWood.com and RichPlanet.net. But even these controlled groups are absolutely tiny in comparison to the Mainstream Media which literally make the entirety of “alternative” media irrelevant. Thus quite literally there is NO alternative media, but instead there are just a handful of truthful people that desperately need you to wake up and help out! Also in this video, I cover some corrections and clarifications from Part 6 regarding the “Meteorite”, Lone Filing Cabinet, and the relevant Kingdome height for WTC rubble comparisons. This was another extensive video on 9/11 which illustrates just how big of an impact it has on nearly all aspects of life and global politics,; and why the controlled opposition groups and PsyOps will continue forever until #FreeEnergy is finally disclosed to the public... Stay Tuned for #911Truth Part 8… CORRECTION: @10:24 Kingdome Cylinder Height is just OVER not under 1/5th the WTC 7 Height. PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh5Bwv78OnP8YtbyfGA View video notes on the Hive blockchain: https://peakd.com/terrorism/@mes/911truth-part-7-how-controlled-opposition-works-alex-jones-cali-dew-fires-flat-earth-more Related Videos: #911Truth: https://peakd.com/truth/@mes/911 #MESScience: https://mes.fm/science-playlist #MESExperiments: https://peakd.com/mesexperiments/@mes/list #AntiGravity: https://peakd.com/antigravity/@mes/series #FreeEnergy: https://mes.fm/freeenergy-playlist . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: Official Website: https://MES.fm Hive: https://peakd.com/@mes Gab: https://gab.ai/matheasysolutions Minds: https://minds.com/matheasysolutions Twitter: https://twitter.com/MathEasySolns Facebook: https://fb.com/MathEasySolutions Pinterest: https://pinterest.com/MathEasySolns Instagram: https://instagram.com/MathEasySolutions Email me: contact@mes.fm Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Try our Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into Conics in Polar Coordinates but this combine a variety of topics together as part of my new video upload strategy. This video covers rotation of polar equations in general, rotation of conic sections in polar coordinates, and finally a brief illustration on how varying the eccentricity affects the shape (and type) of conic section being graphed. I am making this change because my deep dive into the wonderful world of #SuppressedScience and #SuppressedTechnology has been taking up much of my time. By lowering the number of videos but at the same time increasing the amount of content within each video, I am able to save time as well as move further and quicker along my calculus book. So expect some longer and more in-depth videos to come while I continue my epic ongoing research into #FreeEnergy and #AntiGravity! Moving along, in this video I show that rotation of a polar equation is in fact very simple and we can just subtract an angle from the angle of any given function. This results in a rotation counterclockwise of the subtracted angle and about the origin (or polar pole). Either for polar equations in general, or specifically conic sections, this method applies throughout and is much easier to rotate conic sections in polar equations than in the more rigid Cartesian or rectangular coordinates. Varying the eccentricity, or e, of a typical polar equation for a conic section yields some very interesting and surprising results. As e decreases below 1, the shape is an ellipse which is more elongated when e is closer to 1 but becomes more circular as e approaches zero. In fact as e approaches the zero the equation itself approaches zero so a circle is never reached but instead the graph appears to simply reach the exact point of the origin. When e = 1, the function is simply a parabola as expected. When e is greater than 1 the conic is a hyperbola that gets increasingly stretched out as e increases. INTERESTING STUFF! This video is the first of my mathematics videos in this new format, so please let me know what you think by commenting below! Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh5x2M07Ol_vH_HG0Bg View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/conics-in-polar-coordinates-rotation-example-5-and-eccentricity Related Videos: Conics in Polar Coordinates Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0H4OJpJ2gslXVLT8mP-SgJP Conic Sections Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FEUsuxP3KS5DRbidSSGBPL Polar Coordinates Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUFiPLsYw5_Ljd5riOUzjP Conics in Polar Coordinates: Example 2: Ellipse: https://youtu.be/iMy9HN30yIo Polar Coordinates: Example 7: Cardioid: https://youtu.be/rPErcaqNUIY . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I take a break from my epic #AntiGravity Part 6 video which I have been working months on to instead go over a truly amazing formula for determining the Area of a Triangle with knowing only the lengths of the 3 sides. This famous formula is known as Heron’s Formula after the Greek Mathematician Hero (or Heron) of Alexandria as first documented in 60 AD. The formula is stated as such: The Area of a Triangle is the square root of s(s – a)(s – b)(s – c) where a, b, and c are the lengths of the 3 sides of a triangle, and s is the semiperimiter or half the perimeter of the triangle and is equal to (a + b + c)/2. Notice how this formula doesn’t require the height of the triangle or any angle in between the sides. I actually had never encountered this formula before, so I just had to make this video as soon as I learned of it! The derivation method that I used to prove the formula is by using an Algebraic method using the Pythagorean Theorem. Splitting the triangle into two smaller triangles by defining a height as the connecting side, I show that applying the Pythagorean Theorem twice and combined with a LOT of algebra, I can determine the height in terms of the lengths of the sides. Then plugging this height into the common Area = Base x Height / 2 formula, and some reformulation, we get the famous Heron’s Formula! Make sure to watch this video as it is a truly amazing example of how careful and tedious mathematics can eventually lead to a very simple and elegant final formula! #TrulyAmazingStuff And now back to my researching for my #AntiGravity Part 6 video which is shaping up to be one of the most epic and important videos ever made in the history of what we like to call “reality”…. #STAYTUNED! #MyPhD Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh6FcTkdHiZBee2LUsA View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/heron-s-formula-area-of-a-triangle-knowing-lengths-of-3-sides-algebraic-proof Related Videos: Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8 Area of a Triangle - A simple proof that A = b*h/2: http://youtu.be/wUT_KN5pshQ Area of a Triangle Proof - Part 2: When the base is NOT the longest side of the triangle: http://youtu.be/xTVP-HHBP0U . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over an extensive recap on Polar Equations and Polar Coordinates by going over the True-False Quiz found in the end of my Stewart Calculus textbook. This is a 10-Part quiz which involves evaluating each statement to determine if it is true or false, and explaining why and/or providing an example that makes it false. As many of you know by now, I have been busy researching into Suppressed Science, or what I like to call Real Science, so this video serves as a good way to take a break from my research and to get back into the groove of my mathematically challenging tutorials. This video is much more extensive than my usual math tutorials and this is because I plan to spend more time researching and less time producing individual videos. Nonetheless, I will look to cover MUCH MORE math concepts and topics than in previous years by combining them into less videos, but of longer duration. The True-False Quiz I cover involves graphing, differentiating, integrating, and comparing functions in Cartesian, Polar, and Parametric form; including Conic Sections in Polar Coordinates. This is a very extensive and very great way to get caught up on Parametric and Polar Equations in general so make sure to watch this video! MES NOTE: In Question/Statement 10 of the Quiz I also go over a mistake from my earlier video regarding the Hyperbola and its location of the Directrix. The Directrix should be in between the Focus and Center; and NOT as shown in my earlier video: https://peakd.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-hyperbola-proof Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh6V795XVinSZY_2uMw View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/parametric-equations-and-polar-coordinates-true-false-quiz-and-solutions Related Videos: Calculus with Parametric Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0HGaWZ6tZ209j9uYp2f3RLG Parametric Equations and Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0H0AMIZV8HU3LVxSSf_Y9pi Conics in Polar Coordinates: https://www.youtube.com/playlist?list=PLai3U8-WIK0H4OJpJ2gslXVLT8mP-SgJP Area of Polar Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0HEsLZeseN9E9cCgZGYlbe7 Polar Coordinates: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUFiPLsYw5_Ljd5riOUzjP . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over tetrahedral numbers and provide a proof of their formula using mathematical induction. This video was requested by Dominic from Strode College in the UK, and is a good follow-up from my earlier video on the sum of consecutive integers. A tetrahedral number is a number that can be visually represented as 3 sided pyramid, that is a tetrahedron. For example the numbers 1, 4, and 10 can all be visualized as a stack of balls in a tetrahedron pattern. In fact, the tetrahedron is just the sum of consecutive "triangular numbers", which are numbers that can be represented by an equilateral triangle. I cover triangular or triangle numbers as well, and show that their formula is the sum of n consecutive integers. Finally, I use the triangle number formula as well as the principle of mathematical induction to prove the formula for a tetrahedron number. This was the first time I had learned about tetrahedron numbers so this was an interesting topic to cover. If you have any other video requests please let me know and I may cover it! The topics covered as well as their timestamps are listed below. - Introduction: 0:00 - Topics to Cover: 0:45 1. Shoutout to Dominic from Strode College: 1:22 2. Tetrahedral Numbers: 2:26 3. Triangular Numbers: 7:56 - Formula for a Triangular Number: 8:42 - Visualizing the Triangular Number Formula: 10:33 4. Formula for a Tetrahedral Number: 13:51 - Proof by Mathematical Induction: 17:37 Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIibBbOJ0HyyI1sOoEhw?e=VxJ3Yv View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/tetrahedral-numbers-proof Related Videos: Summations Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HyAqJRuDIRPUAvO_3ytXtu Sum of the squares of "n" Consecutive integers - Simple Proof: https://peakd.com/hive-128780/@mes/sum-of-the-squares-of-n-consecutive-integers-proof . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over an extensive video on solving the very interesting and challenging “Problems Plus” examples at the end of my Stewart calculus chapter on Parametric Equations and Polar Coordinates. As explained in my #MESExperiments Introduction video, to free up time to work on my game-changing “alternative” research into free energy, anti-gravity, and all-around real science and real reality, I will be making more condensed and super long mathematics videos like these; instead of the usual short videos for each subsection. This way I will be able to cover more math topics while I upload less videos. In this Problems Plus Examples video, I solve 7 very worth-while problems and they are listed below with their time-stamps to help you navigate through this giant video. These problems combine much of what I covered in my earlier videos on Polar and Parametric Calculus; but also calculus in general as these examples tie in many different mathematical concepts together to solve very abstract and useful problems. I hope you enjoy my new longer and condensed format of mathematical videos, and if you follow along for the ride you may in fact be getting a mathematical education that few if any “Schools” would or can provide! #MESUniversity Intro: 0:00 Question 1: Arc Length of Parametric Curve: 1:21 Question 2: Sketching a Symmetric Curve: 10:40 Question 3: Determining Viewing Rectangle of a Family of Polar Curves: 55:41 Question 4: Spiraling Bugs in Polar Coordinates: 1:23:55 Question 5: Folium of Descartes: 1:55:29 Question 6: Epitrochoid and the Wankel Rotary Engine: 3:21:07 Question 7: Hyperbola Tangent Lines and Equidistant Asymptote Lines: 4:23:50 Stay tuned for my next super long math video! Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh6oci2DcIvOPGSF3Sw View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/parametric-equations-and-polar-coordinates-problems-plus-examples Related Videos: Problems Plus Examples: https://www.youtube.com/playlist?list=PLai3U8-WIK0EmBi_pTs2cBSqfE8oHGHqB Parametric Equations and Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0H0AMIZV8HU3LVxSSf_Y9pi Calculus with Parametric Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0H0AMIZV8HU3LVxSSf_Y9pi Polar Coordinates: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUFiPLsYw5_Ljd5riOUzjP Area of Polar Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0HEsLZeseN9E9cCgZGYlbe7 Conics in Polar Coordinates: https://www.youtube.com/playlist?list=PLai3U8-WIK0H4OJpJ2gslXVLT8mP-SgJP . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In #911Truth Part 11, I have uploaded (with permission) Christian Hampton’s game changing documentary on 9/11, the advanced technology involved, and the continual coverup by controlled opposition “truth” movements. Chris uses his skills as an audio engineer and filmmaker to build upon the work of Dr. Judy Wood, and by focusing on the advanced holographic image projection technology used to mimic large Boeing airliners. Directed (Free) Energy Technology (DEW) is the main type of technology used on 9/11; from creating “plane” illusions to turning 7 large world trade center towers to dust, steering a giant Hurricane Erin off the coast, altering the Earth’s magnetic field, levitating cars, and interfering with nearby cameras and phones; on this day September 11, 2001, 17 years ago, energy, matter, metal, elements, and our very reality was manipulated before our very eyes. It is long overdue that we begin wrapping our minds around what our eyes saw but our minds initially couldn't comprehend... The full description can be viewed here: https://peakd.com/terrorism/@mes/911truth-part-11-feature-documentary-9-11-alchemy-facing-reality-by-wolf-clan-media Wolf Clan Media (WCM) Website: https://wolfclanmedia.org/ WCM YouTube: https://www.youtube.com/user/SyllablezShow The video timestamps are listed below. 1. 0:00 - Introduction by Christian Hampton 2. 1:13 - Predictive Programming? 3. 3:52 - Bill Cooper 9/11 Prediction 4. 4:59 - President Eisenhower Farewell Speech 5. 6:42 - The Energy Issue 6. 7:30 - Controlled Demolitions 7. 8:30 - Missing Rubble & Seismic Impact 8. 10:54 – Black Budget 9. 13:03 – The Aether 10. 14:50 – Earth’s Magnetic Field 11. 15:46 – Hurricane Erin, Tesla Coils, & the day before 9/11 12. 20:17 – Directed Energy Weapons (DEW) 13. 24:35 – Dr. Judy Wood’s Qui Tam Case 14. 24:46 - John Hutchison 15. 27:01 – Awareness Test 16. 27:50 – “Video Fakery” Disinformation 17. 36:47 – Richard Hall’s 3D Radar Analysis 18. 40:59 – The “POD/Drone Theory” Disinfo 19. 43:25 – Dr. Morgan Reynolds Qui Tam Case 20. 51:55 – Contractors: ARA & SAIC 21. 58:04 –3D Projection Technology 22. 1:03:28 – Gelatin & Ace Baker Disinfo 23. 1:09:46 – Real Time Holographic Visualization 24. 1:12:11 – Quasicrystals 25. 1:17:30 – Qui Tam Cases Dismissal & Coverup 26. 1:19:04 – Steven Jones & Cold Fusion Coverup 27. 1:22:14 – Missile Disinfo 28. 1:25:23 – Shanksville: No Plane Debris at Crash Site 29. 1:27:44 – Flight 93 Flight Path 30. 1:32:34 – Flight 77: No Plane Debris at Pentagon 31. 1:34:27 – Pentagon Damage 32. 1:35:34 – Flight 93: Phantom Airplane 33. 1:36:35 – Shiny White Orbs 34. 1:39:46 – Mysterious White Planes 35. 1:41:09 – Disappearing Plane Wings 36. 1:42:42 – Shutter Speed vs Frame Rate 37. 1:46:40 – 1980s Holographic UFOs & Aliens 38. 1:48:12 – Gelatin in Holography 39. 1:49:09 – 2001: A Space Odyssey 40. 1:52:37 – Monolith on 1/1/2001 41. 1:53:49 – Disinfo Agents 42. 1:55:38 – Occult Numerology 43. 1:57:53 – Project Blue Beam 44. 1:59:17 – Holes 45. 2:02:18 – Jumpers 46. 2:04:09 – Star Wars in Iraq & Pentagon 47. 2:05:15 – Warped Columns 48. 2:08:35 – Elevators, Pipes, Concrete Cracking Up 49. 2:11:07 – The “Flash Frame” 50. 2:12:44 – DEW: Side Effects, Camera Glitches, Burger King Outage 51. 2:15:24 – Magnetic Camera Tape 52. 2:17:23 – Power Outages, Unburnt Paper, and Levitation 53. 2:19:08 – Transmutation: Fusion, Fuming 54. 2:22:19 – Flight 93: Cottage in Ruins 55. 2:23:00 – Fuming for Years 56. 2:24:27 – Iron-Rich Spheres: WTC, Crop Circles, Tunguska Event 57. 2:27:11 – Crop Circles: Levitation, High Energy 58. 2:30:24 – Pump Wells: Iron Contamination in Soil 59. 2:33:46 – Flight 93: Energy Effects on Trees 60. 2:35:30 – Space & Perception: Facing Reality 61. 2:38:56 – Strategic Defense Initiative & Project Blue Beam 62. 2:41:24 – Weaponization of Space 63. 2:44:25 – Ashe Conformity Experiments 64. 2:45:53 – Psychological Warfare: War of the Worlds and 9/11 65. 2:51:28 – David Rockefeller: WTC Construction & Destruction 66. 2:52:57 – A Decades Long Conspiracy 67. 2:54:36 – Credits and References Related Videos: #911Truth: https://peakd.com/truth/@mes/911 . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into vectors and this time look at multiplying vectors with the dot product. In my previous videos I discussed adding vectors and multiplying vectors with a scalar. This time I discuss multiplication of two vectors via the dot product, which is defined as multiplying all corresponding vector components and adding them up, thus giving a scalar or real number value; aka not a vector. Further in the video I show that this way of defining vector multiplication has useful applications since it can be reinterpreted in terms of the angle between 2 vectors. I also discuss vector projections, direction cosines, and applications to physics of the dot product. The topics covered as well as their timestamps are listed below. - Introduction: 0:00 - Sections in Calculus Book Chapter: 1:03 - Topics to Cover: 1:46 1. The Dot Product: 2:52 - Definition 1: Dot Product: 3:38 - Example 1: 6:15 - Properties of the Dot Product: 10:49 - Proof of Property 1: 12:42 - Proof of Property 2: 14:00 - Proof of Property 3: 15:59 - Proof of Property 4: 19:15 - Proof of Property 5: 24:00 - Geometric Interpretation of the Dot Product: 24:45 - Definition 2: Dot Product: 28:02 - Example 2: 43:14 - Corollary: 45:46 - Example 3: 46:49 - Perpendicular Vectors: 52:42 - Example 4: 54:47 2. Vector Angles: 56:49 - Direction Angles and Direction Cosines: 1:02:55 - Example 5: 1:17:28 3. Projections: Vector and Scalar Projections: 1:21:27 - Summary of Projections: 1:30:22 - Example 6: 1:33:19 4. Vectors in Physics: 1:39:04 - Example 7: 1:42:33 - Example 8: 1:45:25 Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiahlV-ELBa7AO4z_vw?e=7T9PN5 View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-dot-product View video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GbHgvpTY1mx61nTNO-Msl8 View the full Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ Related Videos: Complex Numbers as Rotation Matrices: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices Vectors: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors 3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In #AntiGravity Part 6, I present the many findings and discoveries made during this past year and a half regarding gyroscopes and all things rotation in general. The discoveries made are on the face of them so simple yet shatter the most overly complicated notions of conventional mainstream physics and science in general. The simple act of rotation defies the very bedrock underneath all of conventional (public) physics. The concept of ‘inertia’ and how it is defined underlies both conventional Newtonian mechanics and Einsteinian Relativity. But the simple act of rotation allows objects to have the magical ability to ‘inertially’ lift upwards, forwards, and even downwards, all without losing additional spin speed and while generating incredible torque over unity yet also behaving ‘massless’. Objects in rotation prove ‘spin angular momentum’, ‘precessional angular momentum’, ‘mass’, ‘inertia’, and hence ‘gravity’ as separate entities are merely illusions; puppets in a puppet show where the puppet master always lay underneath: the Aether. Understanding the true nature of such simple motions as ‘rotation’, which is nothing more than curved linear motion, requires exploring and firmly grasping the true nature of reality itself. From inertia and magnetism to even psychology, technology, and consciousness itself. This video and document, which I am self-appointing as my ‘Doctorate in Philosophy’ or PhD is the precedent for true scientific exploration and the start of an investigation into the true nature of reality, but which has been covered up both by those that rule this world and those that choose to be ruled. Hope you enjoy this epic journey! The topics covered as well as their timestamps are listed below. 1. Important Links and Notes: 1:34 2. Introduction: 5:45 3. Table of Contents: 8:37 4. #MESExperiments Science Experiments Video Series: 55:56 5. General Notes on Scientific Exploration: 58:08 6. Clearing Eric Laithwaite’s Name: 1:06:01 7. Eric Laithwaite’s Unconventional Interpretation of Newtonian Mechanics: 2:25:11 8. Cambridge University’s Bogus Fake Science and Coverup of Gyro Magic: 3:06:30 9. TOP SECRET: The Movie Inception and Spinning Tops: 4:04:00 10. Overview of Conventional Physics Regarding Spinning Tops, Gyroscopes, and Objects in Rotation: 4:07:06 11. Overview of Newton’s Laws and Einstein’s Relativity: 4:52:03 12. Owen Liang’s Gyro Space Top Proves Spin Angular Momentum is an Illusion: 5:28:39 13. INCEPTION IS REAL: Are We Living in a Dream?: 7:06:33 14. Exploring Owen Liang’s Many Unnecessary Accounts: 7:07:36 15. Owen Liang’s Battle with Wikipedia: 9:00:10 16. Francis McCabe and Gyro 1000X+ Torque Over Unity!: 9:19:47 17. Inertial Propulsion and Inertial Lift are Facts of Life: 10:39:17 18. Tippy Tops and Phi Tops Demonstrate Rising with Spin Axis Change: 11:32:58 19. The Amazing and Underappreciated Gyrocompass: 12:33:49 20. The Concept of Inertia and Coherency: 13:30:05 21. Overview of Mainstream Clueless Science Regarding Gyroscopes and Objects in Rotation: 13:42:56 22. SUMMARY OF MY SELF-APPOINTED PHD: 14:25:08 23. MES DUALITY BREAKTHROUGH CONCEPT: 15:50:14 24. AREAS OF FURTHER RESEARCH: 16:32:52 25. Exploring the True Nature of Gravity, Mass, Magnetism, Electricity, Light, and Reality: 18:17:09 26. David LaPoint’s Primer Fields, the Ferrocell, and Ken Wheeler’s Magnetism: 18:27:22 27. THE TO KEY GYROSCOPIC MAGIC IS ‘COUNTERSPACE’: 21:08:36 28. True Magnetism and the Interconnectedness of All Things: 21:45:15 Stay tuned for #AntiGravity Part 7! Video notes: - Word: https://1drv.ms/w/s!As32ynv0LoaIh75UFj1Vekcq1656wA - PDF: https://1drv.ms/b/s!As32ynv0LoaIh-xTV42m_NX2ZDQYIw - HIVE: https://peakd.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept - Backup HIVE link: https://ecency.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept Related Videos: #AntiGravity: https://peakd.com/antigravity/@mes/series #MESExperiments: https://peakd.com/mesexperiments/@mes/list DRAFT #MESExperiments: https://mes.fm/experiments-draft #FreeEnergy: https://mes.fm/freeenergy-playlist #911Truth: https://peakd.com/@mes/911 . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm

In this video, I go over an extremely extensive tutorial on Sequences as part of my new video series on Sequences and Series. Sequences are ordered lists of numbers where each term is categorized by its value and the integer corresponding to its place or order in the list. Series are summations of sequences; which I have shown in my earlier videos in finding areas and hence integrals of functions. The topics I cover in this video include: Sequences and Notation, Definition and Precise Definition, Infinite Limits, Monotonic and Bounded Sequences, as well as the famous Fibonacci Sequence and its relationship with the also very famous Golden Ratio. And at the end of this video I go over a brief mainstream history overview of Leonardo Fibonacci. The Fibonacci sequence is the sequence in which the initial 2 terms are 1 and 1 (or 0 and 1 in modern computing) and then the terms that follow are just the summation of the previous two terms. The limit of the ratio of consecutive Fibonacci terms approach the Golden Ratio (φ) (which is the ratio of any two numbers such that it is the same as the ratio of their summation to the larger number). Fibonacci and Golden numbers/sequences/ratios/spirals/angles/etc. appear very often in all aspect of life from art, nature, design, finance, and even advertising! #AmazingStuff Here is a list of the topics covered and their start times in the video: - @ 0:00 - Intro 1. @ 55:54 - Introduction to Sequences and Series 2. @ 9:25 – Sequences and Notation 3. @ 11:06 - Examples 1 to 3 3. @ 27:30 – Limit of a Sequence 4. @ 33:59 – Definition 1: Limit of a Sequence 5. @ 37:38 – Definition 2: Precise Definition of a Sequence 6. @ 47:33 – Theorem 1 7. @ 52:26 – Definition 3: Infinite Limits of Sequences at Infinity 8. @ 58:17 – Limit Laws for Sequences 9. @ 1:01:39 – Squeeze Theorem for Sequences 10. @ 1:08:10 – Theorem 2 11. @ 1:08:51 - Examples 4 to 7 12. @ 1:18:51 – Theorem 3 13. @ 1:19:48 – Examples 8 to 9 14. @ 1:26:46 – Creating Graphs of Sequences 15. @ 1:29:33 – Example 10 16. @ 1:41:46 – Definition 4: Monotonic Sequences 17. @ 1:43:42 – Examples 11 and 12 18. @ 1:54:08 – Definition 5: Bounded Sequences 19. @ 1:59:31 – Completeness Axiom 20. @ 2:03:43 - Monotonic Sequence Theorem 21. @ 2:09:27 – Example 13: Recursion and Mathematical Induction 22. @ 2:22:27 – Exercise 1: Convergence 23. @ 2:33:20 - Exercise 2: Proof of Theorem 2 24. @ 2:39:41 - Exercise 3: Proof of Theorem 3 25. @ 2:48:23 - Exercise 4: Fibonacci Sequence 26. @ 3:04:17 – Mainstream History of Fibonacci 27. @ 3:12:04 – Overview of Fibonacci Numbers 28. @ 3:17:33 – Overview of the Golden Ratio 29. @ 3:21:53 – Relationship Between Fibonacci Sequence and Golden Ratio 30. @ 3:27:06 – Golden Angle/Spiral and Approximations with Fibonacci Spiral 31. @ 3:31:30 – Fibonacci Numbers and Golden Ratio in Nature, Art, Design, & MORE! #Amazing Stay tuned for my next super long math video! Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh68vMw9e1A75d63O3w View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Related Videos: Limits: https://www.youtube.com/playlist?list=PLFE39AF0D7E0651AC Areas and Summation: https://www.youtube.com/playlist?list=PLai3U8-WIK0E6biFIz0glhAnUwE1qaviI Parametric Equations and Curves: https://www.youtube.com/playlist?list=PLai3U8-WIK0H0AMIZV8HU3LVxSSf_Y9pi . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over a pretty extensive tutorial on infinite series, its definition, and many examples to elaborate in great detail. An infinite series is the summation of a sequence of terms as the terms approach an infinite number of terms and follow from my earlier video on infinite sequences; thus much of the previous video applies to this video. Several important series are covered such as the Geometric series and the Harmonic Series covered. Also, several interesting derivation techniques are utilized such as partial fraction decomposition and the amazing telescoping sum concept. I go over many examples and 2 important exercises from my calculus book (Early Transcendentals by James Stewart) to make the concept of infinite series, and their divergence or convergence, much more clear. The topics and sections covered in this video are listed below with their timestamps: - @ 0:00 - Intro 1. @ 1:34 - Infinite Series and Notation - @ 8:29 - Definition 1 2. @ 13:11 - Example 1: Geometric Series - @ 27:43 - Example 2 - @ 32:49 - Example 3 - @ 39:49 - Example 4 - @ 48:28 - Example 5 3. @ 51:10 - Example 6: Telescoping Sum 4. @ 1:01:31 - Example 7: Harmonic Series 5. @ 1:10:30 - Theorem 1 - @ 1:16:02 - Test for Divergence - @ 1:17:30 - Example 8 6. @ 1:20:35 - Theorem 2 - @ 1:28:15 - Example 9 7. End of Chapter Exercises - @ 1:35:10 - Exercise 1 - @ 1:45:43 - Exercise 2 Download the notes in my video: PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh8gxUmX71Fk9dZ5_wg Excel File: https://1drv.ms/x/s!As32ynv0LoaIh8gwtUIvFFz4HKE0DA View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Related Videos: Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over some mathematics for a change after going all year working on my game changing #AntiGravity Part 6 video. In this video I look at the “Laboratory Project” on Logistic Sequences; note that laboratory projects are very interesting math projects at the end of some sections in my calculus book titled Early Transcendentals by James Stewart. The logistic sequence is defined by the logistic difference equation: Current population = (a constant k) * (previous population)*(1 – previous population) Note that this equation is similar to the logistic differential equation which I covered earlier and is given by the following formula. Population growth = (a constant k) * population (1 – population / carrying capacity K) The logistic sequence involves individual discrete values for the population size and is often preferred for modeling insect populations where mating and death occur periodically. In this project I take a look at modeling the logistic equation using Microsoft Excel spreadsheet to compare how the populations change with changing initial populations and constant k. Interestingly for some values of k there appears to be a leveling off of the population into 1 or more branches. But for other larger values of k the model gives very chaotic and spread out values for the population. The link to the Excel sheet is listed below so make sure to download it and play around with the logistic sequence model! Download the notes in my video: PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh8N8XBPGW5vi1TjDKw Excel File: https://1drv.ms/x/s!As32ynv0LoaIh8N-Iy9FpTaG99wgJA?e=vQl3RQ View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/laboratory-project-logistic-sequences Related Videos: Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Differential Equations: Population Growth: Logistic Equation: https://peakd.com/mathematics/@mes/differential-equations-population-growth-logistic-equation . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video, I go over further into Infinite Sequences and Series and this time look at The Integral Test and Estimating Sums. Often times it is not easy or even possible to formulate a simple function for the n-th partial sum, thus making it very difficult to directly compute the total sum of the series. Thus, other methods are required to first determine if a given series will converge or diverge but without having to do deal directly with the series. I go over one such method, The Integral Test, which geometrically shows that if a series can be written as a function of x and it is decreasing and continuous as x gets large, then the divergence or convergence of the integral of the function itself (i.e. the area under the curve) correlates to the divergence or convergence of the series. A similar approach is shown for estimating the value of a convergent sum and involves considering the minimum and maximum bounds of the remainder in terms of the area or integral of the function as well. This is another in-depth dive into infinite sequences and series so hope you enjoy! - 0:00 - Intro 1. @ 1:30 - Introduction to The Integral Test and Estimate of Sums 2. @ 22:20 - The Integral Test - @ 27:39 - Example 1 - @ 37-20 - Example 2: P-Series - @44:06 - Theorem 1: P-Series - @44:35 - Example 3 - @ 46:56 - Example 4 3. @ 1:00:27 - Estimating the Sum of a Series - @ 1:06:46 - Theorem 2: Remainder Estimate for the Integral Test - @1:08:58 - Example 5 - @1:20:12 - Theorem 3 - @ 1:21:59 - Example 6 4. @ 1:29:31 - Proof of the Integral Test Download Video Notes: - PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh-81BgKD2GACAGwsDA - Excel File: https://1drv.ms/x/s!As32ynv0LoaIh-wyl_SMrkmlNx1E2Q View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Related Videos: Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over a tutorial on using Audacity to remove unwanted background noise in a video or audio file. Audacity is a free and very impressive audio editing piece of software and can be downloaded at https://www.audacityteam.org/. I have had issues with background noise for quite some time and my efforts at removing all practical background noise in the room I record my videos were still not enough. After going over many hours of trial and error and scavenging the internet for audio editing tips, I finally discovered a procedure that worked well for me; and which I have compiled into this video tutorial. Note that when importing MP4 video files into Audacity it automatically separates the M4A audio file but which requires additional libraries to be downloaded and can be done so here: https://manual.audacityteam.org/man/faq_installing_the_ffmpeg_import_export_library.html. The main tools I use is the Audacity “Noise Reduction” and occasional “Amplify” option. It is important to first select the most generic background noise to better remove it from a larger section of the video file. After that I suggest skimming through to find any additional and unique background noise and then remove it from those specific regions of the video. Minimizing the number of edits is key to maintain decent sounding audio and especially speech. Spotting background noise is best done with Noise Cancelling Headphones, so I also recommend obtaining a pair if you are in the business of video or audio production. Later in the video I quickly go over using the Camtasia video editing software to remove the old audio from a video and then add the new fixed audio from Audacity. Camtasia also has a general noise removal option, but it is best done as a second layer to that of Audacity since Camtasia is not as fined tuned or specific as Audacity. I hope you enjoy this tutorial and let me know if it works well for you! Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh-9QBZNbLSsiaRuAOQ View video notes on the Hive blockchain: https://peakd.com/audio/@mes/audacity-background-noise-removal-and-camtasia-video-and-audio-merging Related Videos: GIMP Photoshop Tutorial: http://youtu.be/dSYjD4ov6ao How I Create a Math Easy Solution: http://youtu.be/3Wa98r2Z1Lc Microsoft OneNote Tutorial using Surface Pro Tablet PC: http://youtu.be/YCX8m0Bt-jQ . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go further into the wonderful world of infinite sequences and series and this time look at using comparisons with known series to test if a given series is convergent or divergent. The standard Comparison Test simply shows that if the terms of a series are less than the terms of a known convergent series, such as a p-series or geometric series, than the given series is also convergent; and vice versa if the terms are greater than the terms of a divergent series so too is the given series divergent. I go over many examples and exercises illustrating this test as well as go over a similar Limit Comparison Test. This particular test involves taking the limit as the number of terms approach infinity of the ratio of a given series to a known series; if the ratio approaches a number greater than 0 than both series are either both divergent or both convergent. In later exercises I show that if the ratio approaches 0 and the known series is convergent then so too is the given series; but if the ratio approaches infinity and the known series is divergent than so too is the given series. Many more examples and details are covered in this video so hope you enjoy! The topics and sections covered in this video are listed below with their timestamps: - 0:00 - Intro 1. @ 1:25 - Introduction to the Comparison Tests 2. @ 10:27 - The Comparison Test - @ 25:10 - Example 1 - @ 29:18 - Note #1 - @ 29:55 - Example 2 - @ 36:24 - Note #2 3. @ 38:12 - The Limit Comparison Test - @ 42:35 - Example 3 - @ 45:34 - Example 4 4. @ 55:52 - Estimating Sums - @ 1:00:19 - Example 5 5. @ 1:10:31 - Exercises - @ 1:10:51 - Exercise 1 - @ 1:28:16 - Exercise 2 - @ 1:41:13 - Exercise 3 - @ 1:50:52 - Exercise 4 Download Video Notes: - PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh_ETVsLaaTfgIsAP6g - Excel File: https://1drv.ms/x/s!As32ynv0LoaIh_ESqUCeCXooqaAdqg View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests Related Videos: Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://steemit.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://steemit.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://steemit.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into the wonderful world of Infinite Sequences and Series and this time look at Alternating Series; which are series that alternate from positive and negative terms. When dealing with alternating positive and negative terms it is important to develop specific theorems and methodologies in order to better analyze whether they converge or diverge. Often times it is best to view the odd and even terms of the series separately since they would have the same signs when grouped together. I go over this and much more throughout the video so make sure to follow along! The topics and their timestamps in the video are listed below: - 0:00 - Intro 1. @ 1:28 - Introduction to Alternating Series 2. @ 6:06 - The Alternating Series Test - @ 14:00 - Proof of the Alternating Series Test - @ 27:50 - Example 1 - @ 32:03 - Example 2 - @ 36:28 - Example 3 3. @ 53:43 - Estimating Sums - @ 55:05 - Alternating Series Estimation Theorem - @ 59:27 - Example 4 - @ 1:11:26 - Important Note 4. Exercises - @ 1:11:52 - Exercise 1 - @ 1:48:22 - Exercise 2 - @ 2:08:50 - Proof of the Multiplication Rule for Logarithms - @2:14:05 - Exercise 3 Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_IqKIuB648PDvj1bQ?e=zqe5ZE View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I explore further into Infinite Sequences and Series and this time consider series that have involve the absolute values of a sequence. This involves the concept of “absolute convergence” which I show that it implies convergence in general; although the reverse is not always the case. In this case, the very interesting concept of “conditional convergence” arises which allows for the convergence of a series simply by re-arranging the terms such that the positive and negative terms cancel out; which is only possible if a series is not absolutely convergent. I also go over two specific tests to determine absolute convergence and they are the ratio and root test which are as their names describe. Half of this video is spent on 4 exercises at the end of the video and are a great way to reinforce the concepts introduced in this video. The topics and their timestamps in the video are listed below: - @ 0:00 - Intro 1. @ 1:59 - Absolute Convergence - @ 2:43 - Definition 1 - @ 3:46 - Example 1 - @ 5:53 - Example 2 2. Conditional Convergence - @ 9:22 - Definition 2 - @ 9:54 - Theorem 1 - @ 14:34 - Example 3 3. @ 20:46 - The Ratio Test - @ 44:48 - Example 4 - @ 50:37 - Note on Estimating Sums - @ 51:21 - Example 5 - @ 58:51 - Note on the Ratio Test 4. @ 1:01:25 - The Root Test - @ 1:05:15 - Example 6 5. @ 1:08:46 - Rearrangements 6. Exercises - @ 1:24:52 - Exercise 1 - @ 1:47:01 - Exercise 2: Proof of the Root Test - @ 2:12:30 - Exercise 3 - @ 2:28:40 - Exercise 4 Please note the following list of corrections which I realized during the making of the video: - @ 21:50 – Forgot to include the other condition of part (ii) of the Root Test: lim_(n→∞) |a_n+1/a_n| = ∞. Note that in the video I corrected this @ 36:02 during the proof of part (ii). - @ 2:42:42 - Should’ve written |s_9 – r| = |1 – 5| = 4 less than |a_9| and also should've shown that we can ensure this to be the case regardless of the value of a_9. Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_NXBMhbCTS1dvztYg?e=EZPQ9u View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In #911Truth Part 15 I have uploaded Dr. Judy Wood’s presentation from the 2012 Breakthrough Energy Movement (BEM) Conference in Holland, and which is titled: ‘Evidence of Breakthrough Energy: The Dawn of a New Age’. Judy goes through slide after slide of irrefutable evidence regarding what actually happened on September 11, 2001; which is that the World Trade Center Towers (all 7 of them) were literally turned to dust in a phenomenon before which had not been publicly seen before. Steel, concrete, and even humans turned to dust while leaving very little rubble, seismic impact, or even conventional heat. The fact of the matter is that all this was put on display for all to see, but only Dr. Judy Wood understood what her eyes were telling her instead of the words of others. Judy's investigation into 9/11 goes beyond just the towers and explores evidence that very few minds have even come across; which are listed below: - Cars “toasted”, melted, twisted, levitated, flipped upside down, abnormally peeled, and crumbled. - Steel columns twisted like jelly. - Windows with circular holes on just one glass panel. - A giant Hurricane Erin “coincidentally” being right beside New York City. - Earth’s magnetic field having sharp changes at each major 9/11 event. - Tritium being found without ionization radiation. Interestingly, almost all of the effects present on 9/11 have been replicated by the inventor John Hutchison, “Cold Fusion” discoverers Stanley Pons and Martin Fleischmann, and even natural phenomena such as tornadoes and hurricanes. Dr. Judy Wood has opened the door to what “reality” truly is, and now it is up to us to walk through it… The original video is presented in Global BEM’s YouTube channel as well as on their website. YouTube Video: https://youtu.be/vadSaWyiozg Global BEM Website: https://globalbem.com/dr-judy-wood-evidence-of-breakthrough-energy-on-911/ For more information regarding Dr. Judy Wood’s presentation, the following resources are invaluable: - 500 Page Slide Presentation With Extremely Detailed Notes https://www.slideshare.net/jps2041/dr-judy-wood-911 - PDF Download: https://1drv.ms/b/s!As32ynv0LoaIh_YVV2kTiBzTWwq5WQ?e=VXkjV9 - Dr. Judy Wood’s Recent Interview Going Over Her Presentation: https://youtu.be/p9LMgydKufE To provide further reference, mainly for myself, I have taken screenshots from Dr. Judy Wood’s presentation of the important slides and they are listed in chronological order in the accompanying Hive post: https://peakd.com/freeenergy/@mes/911truth-part-15-feature-presentation-the-dawn-of-a-new-age-by-dr-judy-wood Stay tuned for #911Truth Part 16... Related Videos: ✈️#911Truth Part 11: Feature Documentary: 9/11 Alchemy – Facing Reality by Wolf Clan Media: https://peakd.com/terrorism/@mes/911truth-part-11-feature-documentary-9-11-alchemy-facing-reality-by-wolf-clan-media ✈️#911Truth Part 13: Helicopter Footage of the Twin Towers Turning to Dust (Fast Forward 45X Speed): https://peakd.com/history/@mes/911truth-part-13-helicopter-footage-of-the-twin-towers-turning-to-dust-fast-forward-45x-speed ✈️#911Truth Part 14: Building 7 Turning to Dust for Over 7 Hours (Fast Forward 45X Speed): https://peakd.com/history/@mes/911truth-part-14-building-7-turning-to-dust-for-over-7-hours-fast-forward-45x-speed #911Truth video series: https://peakd.com/truth/@mes/911 #MESScience video series: https://mes.fm/science-playlist #MESExperiments video series: https://peakd.com/mesexperiments/@mes/list #AntiGravity video series: https://peakd.com/antigravity/@mes/series #FreeEnergy video series: https://mes.fm/freeenergy-playlist . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In #MESScience 1, I introduce my latest video series: MES Science Tutorials and continue on my pursuit for understanding true gyroscope physics. Since conventional science doesn’t meet the very high Math Easy Solutions certified (by me) standard, it was inevitable that I create this video series. In the first installment of #MESScience, I take a look at the powerball gyroscope (generically referred to as a gyroscopic exercise tool), then shift gears to the realization that gyroscopes and spinning tops are inverted pendulums without electronic sensors or feedback loops, and then finally finish off with what “precession” may truly be. This is a pretty epic video and it brings my understanding of gyroscopes, which I believe underpins the very foundation of all of science, closer to what is actually going on. This new understanding allows for the possibility of extending conventional physics beyond the mainstream in hopes of quantifying and engineering new applications and discoveries. The topics covered in this video are listed below with their time stamps. - @ 0:00 - Intro - @ 1:09 - Note on MES Hiatus From Uploading Videos - @ 1:55 - Topics to Cover 1. @ 8:38 - Links to Other Series 2. @ 9:04 - Links to #MESScience Series 3. @ 9:29 - Introduction 4. @ 10:54 - Note on the Still Unexplained Gyroscope 5. @ 11:54 - The Powerball Gyroscope 6. @ 22:10 - Mainstream Explanation of the Powerball - @ 44:04 - Quick Summary of Mainstream Powerball Physics 7. @ 46:07 - Recap of my #AntiGravity Part 6 - @ 57:10 - MES REALIZATION: Conventional "Angular Momentum" and Its "Conservation" are at Best "Relative" to Applied Force - @ 1:05:51 - MES REALIZATION: Forced Precession Torque Over-Unity Depends on Amount of Forced Supplied - @ 1:12:23 - Quick Summary of #AntiGravity Part 6 Findings 8. @ 1:14:15 - Effects of Rolling, Spin Speed, and Added Weight on Gyros and Spinning Tops - @ 1:51:21 - Eric Laithwaite Had Unwittingly Demonstrated the Basic Mechanics of a Powerball in 1974 - @ 1:58:06 - MES REALIZATION: Precession of Spinning Tops Should Not Be Confused With Rolling Trajectory - @ 2:04:06 - Quick Summary of Effects of Rolling, Spin Speed, and Added Weight 9. @ 2:05:45 - MES Breakdown of and Experiments with a Gyro Powerball - @ 2:25:58 - Quick Summaries of MES Powerball Breakdown and MES Powerball Experiments 10. @ 2:28:50 - MES BOMBSHELL DISCOVERY: Added Weight Can Prevent Rising on Low Friction Surfaces - @ 2:32:44 - Does Preventing Centrifugal Precession Condense into a Spiraling Upwards #Vortex Precession? 11. @ 2:36: 17 - MES BOMBSHELL REALIZATION: Gyroscopes and Spinning Tops are Inverted Pendulums Without Electronic Sensors - @ 2:43:50 - Mathieu's Equation and My Previous Lifetime?? #MESDuality - @ 2:49:25 - MES Custom Gimballed Gyroscope Inverted Pendulum 12. @ 3:05:05 - Quick Summary of Recent Bombshell Gyro Realizations 13. @ 3:08:35 - I THINK I SOLVED PRECESSION! - @ 3:33:20 - MES BOMBSHELL REALIZATION: Precession is Merely Gyro Orientation Change Due to Applied Torque - The Conventional View of Precession is Wrong! - MES BOMBSHELL REALIZATIONS: - @ 3:39:22 - Precession is a Torque and Moves the Ice Block - @ 3:41:58 - Precession is Always “Forced” - @ 3:56:05 - Spin Angular Momentum 'Absorbs' and Re-Directs Torque - The Most Accurate Representation of Gyroscopic Anti-Gravity Motions - @ 4:03:03 - Quick Summary of MES True Precession Stay Tuned for #MESScience 2… PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIh_xmfIOk-PgRi0gdyg?e=8KXRlZ Hive Notes: https://peakd.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums Related Videos: #MESExperiments video series: https://peakd.com/experiments/@mes/list DRAFT #MESExperiments video series: https://mes.fm/experiments-draft #MESScience video series: https://mes.fm/science-playlist #AntiGravity video series: https://peakd.com/antigravity/@mes/series #FreeEnergy video series: https://mes.fm/freeenergy-playlist . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into infinite sequences and series and this time look at the Power Series which is a series in which a power function is being summed up. The power series has many useful applications such as the vibrations of a drum as in the Bessel function and has to do with the particular behavior of the power series to oscillate in large amplitudes initially and dissipate with distance and/or time. I go over many examples, exercises, and proofs in yet another epic long math video, so follow along to learn more! The topics and their timestamps in the video are listed below: - @ 0:00 - Intro 1. @ 1:54 - Power Series - @ 9:41 - Note on Trigonometric Series and the Fourier Series - @ 10:40 - Example 1 - @ 14:33 – Note on Factorials - @ 15:23 - Example 2 2. @ 24:31 - Note on Bessel Functions - @ 25:49 - Example 3: Bessel Function of Order 0 3. @ 49:41 - Theorem 1 - @ 55:54 - Summary of Previous Examples - @ 58:35 - Example 4 - @ 1:07:22 - Example 5 4. Exercises - @ 1:17:18 - Exercise 1: Bessel Function of Order 1 - @ 1:34:44 - Exercise 2: Proof of Theorem 1 - @ 1:35:53 - Theorem 2 - @ 1:49:24 - Theorem 3 - @ 1:53:48 - Completeness Axiom Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_ZL0WExNvIL4J0s2Q?e=GkZR2u View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-power-series Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests Infinite Sequences and Series: Strategy for Testing Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into infinite series and sequences and this time summarize an 8-step strategy for testing any given series for convergence or divergence. The strategy is not to be followed in order as that would be time consuming but rather to compare the type of series being dealt with the steps in the strategy. Since in my earlier videos I went over alternating series, geometric series, p-series, absolute series, and series with factorials or incrementally increasing powers (i.e. to the power of n), this video serves as a good summary and game plan for deciding how to go about tackling each type of series being presented. I also over 6 quick examples illustrating which test to perform. The topics and their timestamps in the video are listed below: - @ 0:00 - Intro 1. @ 0:52 - Strategy for Testing Series - @ 1:32 - Step 1: P-Series. - @ 2:14 - Step 2: Geometric Series - @ 3:19 - Step 3: Comparison Tests - @ 7:32 - Step 4: Test for Divergence - @ 8:18 - Step 5: Alternating Series Test - @ 10:25 - Step 6: Ratio Test - @ 16:00 - Step 7: Root Test - @ 16:48 - Step 8: The Integral Test 2. Examples - @ 18:40 - Example 1: Test for Divergence - @ 20:34 - Example 2: Limit Comparison Test - @ 23:16 - Example 3: The Integral Test (or Ratio Test) - @ 26:54 - Example 4: Alternating Series Test - @ 27:20 - Example 5: Ratio Test - @ 28:38 - Example 6: Comparison Test Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_YMrfZ1-cLDGKPt7Q?e=Q56IeX View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over a very interesting concept and that is in interpreting complex or imaginary numbers as and equivalent rotational matrix form. This concept was brought to my attention by Lori Gardi from the Fractal Woman YouTube channel: https://www.youtube.com/user/FractalWoman/search?query=complex%20numbers Rewriting complex numbers as rotation matrices allows for looking at complex numbers from a different perspective, and this case that perspective is in simply rotating vectors. The imaginary number "i" is thus viewed as simply a 90 degree rotation when viewed as a rotation matrix. I also rewrite the famous Euler's formula into a rotation matrix form and that turns out to just be a 180 degree rotation. Since complex numbers appear often in mathematics and physics, it may be insightful to reinterpret such complex equations as simply vector rotations! Note that while I only covered 2D rotation matrices, a similar concept can be applied to 3D rotation matrices, and which I may cover in future videos so stay tuned. The topics covered as well as their timestamps are listed below. - Introduction: 0:00 - Topics to Cover: 0:41 1. Fractal Woman and Other References: 1:28 2. Rotating a Vector: 3:01 3. Matrix Multiplication: 12:47 4. Matrix Addition: 25:29 5. Rotation Matrix: 27:13 6. Complex Numbers: 36:58 7. Relating the Complex Plane with the Rotation Matrix: 43:30 - Complex Numbers as Vector Rotations: 57:12 - Reinterpreting Euler's Formula: 1:10:02 - Applications to Other Fields: 1:20:51 Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiaFTZDoqo7BXtv_aCQ?e=vfGRLy View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices View video sections playlist: https://www.youtube.com/playlist?https://www.youtube.com/playlist?list=PLai3U8-WIK0E6x0dZEAx77Kqv7LLxYdbx View the full Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ Related Videos: Vectors: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors 3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into infinite sequences and series; and in particular expand upon my earlier video on power series. This time I show how we can represent many functions as power series as well as go over a theorem that allows for differentiation and integration of power series term by term. This theorem, as well as this overall concept of representing functions as power series is one of the most important concepts in all of science because it allows for much easier computation of many different functions. In fact many computers and calculators use this technique behind the scenes since they are often much easier to deal with than trying to find exact integrals or derivatives which many times are not simple. In this video, and at the second half of this video, I also go over an extensive 2 hour long proof of the theorem for term-by-term differentiation and integration of power series. Although this proof is extensive, after searching all over online for the proof, I believe my proof may in fact be the most clear and understandable. Let me know if it helps! The topics covered in this video are listed below with their time stamps. - @ 0:00 - Intro 1. @ 2:24 - Representation of Functions as Power Series - @ 3:17 - Equation 1 - @ 10:48 - Example 1 - @ 26:35 - Example 2 - @ 30:16 - Example 3 2. - @ 37:06 - Differentiation and Integration of Power Series - @ 38:09 - Theorem 1: Term by Term Differentiation and Integration - @ 50:50 - Example 4 - @ 54:18 - Example 5 - @ 58:36 - Example 6 - @ 1:11:02 - Example 7: Gregory's Series - @ 1:25:59 - Leibniz Formula for π - @ 1:30:30 - Example 8 3. Exercises - @ 2:11:57 - Exercise 1 - @ 1:55:34 - Exercise 2 - @ 2:26:11 - Exercise 3: Proof of Theorem 1 - @ 2:32:29 - Lemma - @ 2:34:58 - Lemma Derivative Proof - @ 3:01:16 - Lemma Integral Proof - @ 3:08:59 - Theorem 1 Using z = (x - a) - @ 3:12:41 - Theorem 1a Derivative Proof - @ 3:24:59 - Brief Overview of the Binomial Theorem - @ 3:57:33 - Theorem 1b Integral Proof Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIh_17muwohTwCBpQ45Q?e=pldHwj View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests Infinite Sequences and Series: Strategy for Testing Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series Infinite Sequences and Series: Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-power-series . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over further into the wonderful world of sequences and series, and this time go over Taylor and Maclaurin series. These series are precise and ingenious ways of approximating many functions by only knowing all of the function’s derivatives at a specific value; in the case of the Maclaurin series it is as x = 0. These series can also be used to integrate functions that don’t have elementary antiderivatives and thus are utilized in many computers and calculators as the basis behind many types of calculations. Interestingly, it was this very concept of Taylor and Maclaurin series that first gave me the idea of creating a YouTube channel to better learn and organize my learning of mathematics and beyond. Taylor and Maclaurin series, as well as infinite series in general, require the coming together of many mathematical concepts and thus I had then realized while still in University that I need to fully refine my mathematical knowledge. And this particular video is the perfect illustration of such a culmination of so many various mathematical concepts. The topics covered in this video are listed below with their time stamps. - @ 0:00 - Intro - @ 1:14 – MES Note on Taylor and Maclaurin Series 1. @ 5:16 - Introduction to Taylor and Maclaurin Series - @ 21:38 - Theorem 1 - @ 22:41 - Taylor Series - @ 24:43 - Maclaurin Series - @ 26:25 - Historical Note on Taylor and Maclaurin - @ 27:37 - Example 1 2. @ 35:00 - When is a Function Equal to the Sum of Its Taylor Series? - @ 36:18 - Taylor Polynomials - @ 49:01 - Theorem 2 - @ 49:49 - Taylor's Inequality - @ 51:36 - Proof for n = 1 - @ 1:14:09 - Equation 1 - @ 1:15:03 - Note on Alternatives to Taylor's Inequality - @ 1:17:41 - Example 2: e^x - @ 1:30:11 - Historical Note on Calculating the Number e - @ 1:31:15 - Example 3 - @ 1:35:04 - Example 4: sin x - @ 1:48:33 - Example 5: cos x - @ 1:55:31 - Historical Note on Maclaurin Series - @ 1:56:23 - Example 6: x cos x - @ 1:58:04 - Example 7 - @ 2:11:23 - Note on Power Series 3. @ 2:12:30 - Comparing Taylor and Maclaurin Series 4. @ 2:15:44 - Example 8: (1 + x)^k - @ 2:32:26 - The Binomial Series - @ 2:45:43 - Example 9 5. @ 2:59:09 - Table 1: Important Maclaurin Series and Their Radii of Convergence 6. @ 3:04:12 - Integrating Functions that Don't Have Elementary Antiderivatives - @ 3:09:35 - Example 10: e^(-x^2) 7. @ 3:28:43 - Example 11: Evaluating Limits 8. @ 3:38:26 - Multiplication and Division of Power Series - @ 3:39:33 - Example 12 9. Exercises - @ 3:57:28 - Exercise 1: A Function Not Equal to its Maclaurin Series - @ 4:17:05 - Exercise 2: Proof of Taylor's Inequality for n = 2 - @ 4:37:01 - Exercise 3: Proof of Binomial Series View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-taylor-and-maclaurin-series Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiIFz-mgCcPKh_6t2Sg?e=vhxEFP Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests Infinite Sequences and Series: Strategy for Testing Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series Infinite Sequences and Series: Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-power-series Infinite Sequences and Series: Representations of Functions as Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I expand upon my previous video on Taylor and Maclaurin series by going over a few alternative forms of the Remainder Term. Recall that the Remainder Term is just that, a remainder term, and is what is left over when approximating a function with its Taylor series. In the previous video I showed that we can use Taylor’s Inequality to determine the remainder term, but in this video, I show that we can also use an integral form as well as a form called Lagrange’s form of the remainder term. I prove both of these forms of the remainder as well as going over some examples to compare with the previous video’s use of Taylor’s Inequality. The topics covered in this video are listed below with their time stamps. - @ 0:00 - Intro 1. @ 2:36 - Recap on Taylor and Maclaurin Series 2. @ 8:26 - Formulas for the Remainder Term in Taylor Series 3. @ 11:32 - Theorem 1: Integral Form of the Remainder Term - @ 40:48 - Recap of Example 4 from Previous Video - @ 43:44 - Example 1: sin x - @ 52:24 - Absolute Value Integral Property 4. @ 1:08:42 - Lagrange's Form of the Remainder Term - @ 1:13:38 - Theorem 2: Weighted Mean Value Theorem for Integrals - @ 1:28:20 - Theorem 3: Lagrange's Form of the Remainder Term - @ 1:40:36 - Example 2: sin x - @ 1:44:35 - Example 3: x^(1/3) Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiIMceHV2_A1JziAI4A?e=E7aeMn View video notes on the Hive blockchain: https://peakd.com/@mes/infinite-sequences-and-series-formulas-for-the-remainder-term-in-taylor-series Related Videos: Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz Infinite Sequences: Limits, Squeeze Theorem, Fibonacci Sequence & Golden Ratio + MORE: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more Infinite Series: Definition, Examples, Geometric Series, Harmonics Series, Telescoping Sum + MORE: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more Infinite Sequences and Series: The Integral Test and Estimate of Sums: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests Infinite Sequences and Series: Absolute Convergence and the Ratio/Root Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-absolute-convergence-and-the-ratio-root-tests Infinite Sequences and Series: Strategy for Testing Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series Infinite Sequences and Series: Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-power-series Infinite Sequences and Series: Representations of Functions as Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series Infinite Sequences and Series: Taylor and Maclaurin Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-taylor-and-maclaurin-series . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps