Premium Only Content

Infinite Sequences and Series: Multiplication and Division of Power Series
In this video I go over a very in-depth tutorial on division and multiplication of power series just as we would for polynomials. In my earlier video I had gone over an example on multiplying and dividing power series using the basic polynomial expansion and polynomial long division techniques, but I had not discussed the reasoning for why it works. In this video, however, I derive the power series that results from multiplying two power series and show that the resulting coefficients involve a “convolution” summation that can also be obtained in the same way had we just multiplied the power series as if they were polynomials.
The division of power series is effectively just the reverse of multiplication and thus to divide a power series is the same as figuring out which power series multiplies to get the numerator. I show that polynomial long division and double checking using multiplication yield the same result as expected.
Also in this video, I go over briefly on Euler’s identity and imaginary numbers to illustrate, in Example 3, that if we consider imaginary numbers then we need to account for when the denominator of a power series is equal to zero in the imaginary or complex set of numbers. This particular example was in regard to obtaining the power series of hyperbolic cosecant or csch.
The topics covered in this video are listed below with their time stamps.
- @ 0:00 - Intro
- @ 1:14 - Topics to Cover
1. @ 2:19 - Recap on Power Series
- @ 5:54 - Taylor and Maclaurin Series
2. @ 6:59 - Recap on Earlier Example on Multiplication and Division of Power Series
3. @ 18:21 - Multiplication of Power Series
- @ 49:43 - Example 1: e^x/(1-x)
- @ 1:03:36 - Example 2: sin^2(x)
4. @ 1:27:37 - Division of Power Series
- @ 1:34:04 - Example 3: csch(x)
- @ 1:53:20 - Imaginary Numbers and Euler's Identity
- @ 2:02:08 - Double Checking Division By Multiplication
Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiLx8qaLYwrw4kpWAjg?e=uD08dM
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-multiplication-and-division-of-power-series
Related Videos:
Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
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"
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
Leibniz’s Rule: Generalization of the Product Rule for Derivatives: https://peakd.com/hive-128780/@mes/leibniz-general-product-rule-for-derivatives
Polynomial Long Division - In depth Look on why it works! http://youtu.be/E1H584xJS_Y
Foil Method - Simple Proof and Quick Alternative Method: http://youtu.be/tmj_r94D6wQ .
------------------------------------------------------
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
-
3:25:53
Math Easy Solutions
19 days ago $0.18 earnedMES Livestream 101: How Controlled Opposition Works
4621 -
LIVE
megimu32
2 hours agoOTS: Pimp My Ride LOL - Insane Mods & Car Stereo Nostalgia
126 watching -
1:29:46
Glenn Greenwald
4 hours agoIsraeli Official Caught in Pedophile Sting Operation Allowed to Flee; Israeli Data: 83% of the Dead in Gaza are Civilians; Ukrainian Man Arrested over Nord Stream Explosions | SYSTEM UPDATE #503
68.8K49 -
15:27
robbijan
5 hours ago $1.12 earnedThe Emperor’s New Labubu & The Spiritual War Behind Everything
21.5K20 -
16:40
T-SPLY
1 day agoLiberal Lawmaker Gets Busted Lying - D.H.S Brings Receipts!
10.8K9 -
LIVE
SavageJayGatsby
2 hours agoMario Party with Friends | Goal to 100 Followers | 300 Weekly Goal for Spicy Bites!
48 watching -
LIVE
Mally_Mouse
4 hours agoLet's Play!! - Mario Party Jamboree w/FRIENDS
52 watching -
26:53
Stephen Gardner
2 hours ago🔥Dems NEW PLOT Against Trump EXPOSED!
9.38K17 -
1:00:55
The Mel K Show
5 hours agoMel K & Patrick Byrne | Deep Rig Exposed: Tracking Election Fraud's Global Tentacles | 8-21-25
16.7K2 -
1:07:38
BonginoReport
5 hours agoThe Po-Po President Picks Up The Night Shift! - Nightly Scroll w/ Hayley Caronia (Ep.117)
91.1K48