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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:

- @ 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

- @ 24:31 - Note on Bessel Functions
- @ 25:49 - Example 3: Bessel Function of Order 0

- @ 49:41 - Theorem 1
- @ 55:54 - Summary of Previous Examples
- @ 58:35 - Example 4
- @ 1:07:22 - Example 5

- 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 .

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