Infinite Sequences and Series: Power Series

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. @ 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

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