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

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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:

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