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

Key Concepts

Power Series

A power series is an infinite series in the form ?a?(x - c)? where c is the center of the series. Analyzing power series involves determining for which values of x the series converges. Even when the series appears in a slightly different form, it can often be rewritten to reveal its center and structure.

Rewriting into Standard Form

Sometimes a series is given in a non-standard form, such as using an expression like (4x + 1)?, which can be rewritten as [4(x + 1/4)]?. This form makes it easier to identify the center of the series (in this case, x = -1/4) and to relate the variable part to a standard power series in (x - c).

Radius of Convergence

The radius of convergence is the distance from the center of the series within which the series converges. It is usually found by applying convergence tests, like the ratio test, to the general term of the series. When the series is recast in a standard form, the inequality obtained (e.g., |4(x + 1/4)| < 1) can be used to calculate the radius (here, 1/4).

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges. After determining the radius of convergence, one translates the condition (|x + 1/4| < 1/4) into an interval, here (-1/2, 0). The behavior at the endpoints must be examined separately to ascertain whether the series converges there.

Absolute Convergence and Endpoint Analysis

Absolute convergence is a condition where the series of absolute values converges. At the boundaries, even if the ratio test is inconclusive, convergence tests (such as the p-series test) can be used to determine if the series converges. In this context, evaluating the series at the endpoints confirms absolute convergence, which means that the endpoints are included in the interval of convergence.

Transcript

00:01 In this question, we are asked to find the radius of convergence and the interval of convergence for the following series.

00:08 First, the first thing we need to do is to rewrite this series in the form of a power series.

00:17 Recall that the general form of power series is ck times x minus a to the k, right? and the important thing to note here is that the coefficient in front of x x, is equal to 1.

00:35 So we want to do same for our series.

00:39 Let's factor out four out with the brackets, out of parentheses, we are going to get 4 to the n times x plus 1 or 4 to the n divided by n squared.

00:53 Now our series looks like a power series.

00:58 And to find its radius of convergence and the interval of convergence, we are going to use the ratio test.

01:10 By the ratio test, we first need to calculate the limit of a .n plus 1 over a .n as n goes to infinity.

01:20 This is equal to the limit of 4 to the n plus 1 times x plus 1 over 4 to the n plus 1 squared times the reciprocal of a .n, which is going be n squared over 4 to the n times x plus 1 over 4 to the n now this is going to be the limit of 4 times x plus 1 over 4 times n squared over n plus 1 squared and since the limit of n squared over n plus 1 squared is equal to 1 we are going to get 4 times absolute value of x plus 1 over 4 and by the ratio test for the series to converge, we want this limit to be less than 1.

02:33 We can rewrite this as absolute value of x plus 1 over 4 is less than 1 over 4.

02:41 And this immediately gives us the radius of convergence of the series, which is going to be 1 over 4...

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