Find the radius of convergence and interval of convergence...

Key Concepts

Power Series

A power series is an infinite series of the form ? a?(x - c)? where a? represents the coefficient of the nth term and c is the center of the series. This representation is fundamental in analysis for expressing functions as sums of polynomial terms and is useful in calculus and differential equations.

Radius of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges absolutely. It is a critical concept because it defines the interval in which the power series accurately represents a function, and it is usually determined using tests like the ratio test or the root test.

Interval of Convergence

The interval of convergence refers to the complete set of x-values for which a power series converges. This includes not only the open interval defined by the radius of convergence but also requires a separate analysis of the endpoints to determine whether the series converges or diverges there.

Convergence Tests

Convergence tests such as the ratio test or the root test are essential tools in determining the behavior of the power series. These tests analyze the limit of the sequence of terms or ratios of successive terms to establish the radius of convergence and to check for absolute convergence within that radius, while additional tests may be used for evaluating convergence at the endpoints.

Transcript

00:01 Okay, first step is to use the ratio test.

00:03 So limit as in goes to infinity, x to the n plus 1, over 2 times n plus 1 minus 1, over x to the n, over 2n minus 1.

00:18 So that's the limit, as in goes to infinity, x to the n plus 1 over 2n plus 1 times 2n minus 1 over x to the end.

00:31 That's the limit as in goes to infinity x to the n plus 1 over x to the n 2n minus 1 over 2n plus 1 so that's equal to x times the limit as in goes to infinity 2n minus 1 over 2n plus 1 okay so that's infinity over infinity and if you lopetal's rule you get 2 over 2 which is equal to 1 so that limit is 1 so x less 1 1.

01:13 So the radius of convergence is 1.

01:20 Now we have to check the end points.

01:26 Let x equal minus 1, and the series turns into minus 1 to the n over 2n minus 1.

01:37 It's an alternating series.

01:39 So we'll try the alternating series test.

01:42 First take the limit, and that's 0.

01:48 Okay, n plus 1 is bigger than in so two times in plus one is bigger than two times in so two times in plus one minus one is bigger than two n minus one so one over two in plus one minus one is less than one over two n minus one so it's decreasing therefore convergent by the alternating series test and now let x equal one so it turns to 1 to the n over 2n minus 1, or 1 over 2n minus 1.

02:44 Okay, i'm going to say this is like 1 over n, the divergent harmonic series...

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