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

Key Concepts

Endpoint Convergence Testing

After determining the radius of convergence, each endpoint of the interval must be tested individually because the convergence behavior at these points may be different from that within the open interval. Techniques like the Alternating Series Test or other relevant tests are used to analyze the behavior at the endpoints.

Convergence Tests

A variety of convergence tests—including the Ratio Test, Root Test, and Alternating Series Test—are employed to determine the behavior of series. These tests help identify whether a series converges absolutely, conditionally, or diverges, which is critical when working with power series.

Radius of Convergence

The radius of convergence is the distance from the center of the series (c) within which the power series converges absolutely. It can be determined using tests such as the Ratio Test or the Root Test by analyzing the behavior of the coefficients.

Power Series

A power series is an infinite series expressed in the form ? a? (x - c)?, which represents a function as an infinite sum of terms involving powers of (x - c). The convergence of the series depends on the value of x and the coefficients a?.

Interval of Convergence

The interval of convergence includes all x-values for which the power series converges. This interval is centered at c and spans from (c - R) to (c + R), with the endpoints typically requiring separate analysis to determine if convergence holds there.

Transcript

00:01 We want to find the radius and the interval of convergence for this power series, so we'll use the ratio test.

00:08 So you take the limit as in goes to infinity of the n plus first term.

00:13 And since we're taking the absolute value of this ratio that we're going to write here, the n plus first term divided by the nth term, we don't have to put the minus 1 to the n in there because they're going to cancel that anyway.

00:28 X to the n plus 1 over the cube root of n plus 1.

00:33 Over x to the n over the cube root of n.

00:39 Okay, so invert and multiply the bottom one.

00:43 Limit as in goes to infinity, x to the n plus one, cube root of n plus one, times cube root of n over x to the n.

00:59 Okay, so now i'm gonna rewrite it and put the similar pieces together, x to the n plus one over x to the n, cube root of n over the cube root of n plus one.

01:16 All right, you have n plus 1, x is on the top, and n of them on the bottom that leaves one of them on the top.

01:24 And then this is the cube root of n over n plus 1.

01:30 All right, so the limit of the product is the product of the limit.

01:33 So we take the limit of x, so we get x, and then we take the limit as in goes to infinity of the cube root of n over n plus 1.

01:45 So in fact, we can just go all the way into the cube root and take the limit.

01:57 Okay, so that's infinity over infinity.

01:59 If you love patel's rule, you get 1 over 1, so that limit is 1.

02:02 So you get the absolute value of x.

02:05 And then you set it less than 1 because the ratio test says, if that ratio is less than 1, this series is convergence.

02:15 Okay, so that means our radius of convergence is 1.

02:22 All right, so we know for sure that it's convergent between 1 and negative 1...

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