Review Question 9: Derivative and Integral of Power Series

In this video I recap on term-by-term differentiation and integration of power series. This theorem, which I proved in my earlier video, states that the derivative of a power series is equivalent to first taking the derivative of each term and then summing it up. Likewise this is true for integrating a power series as well. In both cases, the radius of convergence is the same as that of the original power series. Note that the interval of convergence need not be the same for either case. Differentiating power series term-by-term has an important application in solving differential equations so make sure you fully grasp this concept!

The timestamps of key parts of the video are listed below:

Question 9: 0:00
(a) Derivative of Power Series: 0:32
(b) Integral of Power Series: 3:59
Radius of convergence of derivative and integral is R: 6:06
Rewriting the derivative and integral equations: 6:27
Note 1: Infinite sum vs finite sums: 7:39
Note 2: Interval of convergence may change: 8:26
Note 3: Application in solving differential equations: 8:58

This video was taken from my earlier video listed below:

Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L