Decide if the statements are true or false. Give an...

Key Concepts

Power Series

A power series is an infinite series of the form ? a?(x - c)?, where each term involves a power of (x - c) and a coefficient a?. Understanding power series is crucial because their convergence properties depend heavily on the value of x relative to the center c.

Interval of Convergence

The interval of convergence of a power series is the set of all x-values for which the series converges. This interval typically takes the form (c - R, c + R), where R is the radius of convergence, but the behavior at the endpoints must be determined separately.

Endpoint Convergence

Convergence at the endpoints of the interval of convergence is not automatically determined by the behavior within the radius of convergence. Each endpoint must be tested individually because the series may converge at one endpoint while diverging at the other.

Absolute vs. Conditional Convergence

A series converges absolutely if the series of absolute values converges, and conditionally if it converges only when the absolute values do not. This concept is important at endpoints, where a power series might exhibit conditional convergence at one endpoint but divergence at the other.

Transcript

00:01 In this problem, we need to determine whether a given statement is true or false.

00:07 Now, in this problem, the given statement is that if a power series converges at one endpoint of its interval of convergence, then it must converge at the other.

00:17 Now, in order to determine whether this is true or false, consider the series summation n is equal to 1 to infinity minus 1 to the power n divided by n times x to the power n.

00:30 This is a power series and the nth coefficient cn will be minus 1 to the power n divided by n.

00:41 And so by replacing n by n plus 1, we get the n plus 1th coefficient to be equal to minus 1 to the power n plus 1 divided by n plus 1.

00:51 Now let us determine the radius of convergence.

00:55 Now the radius of convergence can be obtained as limit n.

01:01 N tends to infinity, modulus of cn divided by the modulus of cn plus 1.

01:12 Now that will be equal to the limit as n tends to infinity of the modulus of cn is the modulus of minus 1 to the power n by n and this will just be 1 divided by n and the modulus of cn plus 1 will be the modulus of minus 1 to the power n plus 1 by n plus 1.

01:34 And that will become 1 by n plus 1.

01:37 So this will be equal to the limit as n tends to infinity of n plus 1 divided by n.

01:47 So this can be written as limit n tends to infinity 1 plus 1 by n.

01:54 And since 1 by n tends to 0 as n tends to infinity, this will be equal to 1 plus 0, which is 1.

02:00 So the radius of convergence is 1...

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