Infinite Sequences and Series: Representations of Functions as Power Series

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:

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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 .

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