Review Question 8: Power Series, Radius and Interval of Convergence

In this video I recap on power series and the conditions that it converges, that is, its radius and interval of convergence. A power series has terms of the form: constant * (x - a)ⁿ. There are only three possibilities for its convergence: Either it converges only for x = a, or converges for all x, or it converges for |x - a| is less than the radius of convergence R. In the first case, the radius of convergence is considered as R = 0, and the second case it is R = infinity. The interval of convergence has 4 possibilities, since at |x - a| = R, the series can converge or diverge. Thus the interval of convergence is either (a - R, a + R), (a - R, a + R], [a - R, a + R), or [a - R, a + R]. I also graphically show this in the video.

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

• Question 8: 0:00
• (a) General form of a power series: 0:20
• (b) Radius of convergence of a power series: 1:11
  • Summary for the radius of convergence: 6:28
• (c) Interval of convergence of a power series: 7:40

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

Related Videos:

Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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