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

Key Concepts

Endpoint Analysis

Endpoint analysis involves testing the convergence of the power series at the boundaries of the interval determined by the radius of convergence. Since the Ratio Test is inconclusive when the limit equals one, other convergence tests, such as the Alternating Series Test or the p-Series Comparison, are employed to determine whether the series converges or diverges at these points.

Ratio Test

The Ratio Test is a method used to determine the convergence of an infinite series by evaluating the limit of the absolute value of the ratio of successive terms. In the context of power series, it helps isolate the variable x and facilitates finding the radius of convergence by ensuring that the ratio is less than one within the interval of convergence.

Power Series

A power series is an infinite series of the form ?a?(x - c)?, where c is the center and a? are the coefficients. Understanding how a function can be represented locally by a power series is crucial for analyzing its convergence properties and performing manipulations such as differentiation and integration term-by-term.

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 determined by applying tests, such as the Ratio Test or the Root Test, which relate the behavior of the sequence of coefficients to the variable part of the series, ensuring that |x-c| remains less than the radius for convergence.

Interval of Convergence

The interval of convergence is the set of all x values for which the power series converges. While the radius of convergence provides a bound for absolute convergence, the interval must be checked at the endpoint values separately since convergence or divergence at the endpoints can require additional analysis using specific convergence tests.

Transcript

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

00:08 To do that, we are going to use a ratio test.

00:14 By the ratio test, we first need to calculate the limit of a n plus 1 over a .n as n goes to infinity.

00:25 In our case, this is going to be n plus 1 times x minus 4 to the n plus 1 over n plus 1, multiplied by the reciprocal of a n which is going to be n q plus 1 divided by n times x minus 4 to the n this is equal to the limit of absolute value of n plus 1 times n q plus 1 over n plus 1 q plus 1 times n plus 1 times n multiplied by x minus 4.

01:28 Now recall that to calculate the limit of a rational function, and here we have a rational function, we just need to look at the coefficients in front of the highest powers of n in the numerator and the denominator.

01:51 So the highest power of n in the numerator is n to the 4, and the coefficient is equal to 1.

01:59 The highest power of n in the denominator is n to the 4.

02:02 And the coefficient is also equal to 1.

02:07 Therefore, the ratio is going to be 1 over 1, and this whole thing, this rational function goes to 1 as n goes to infinity.

02:17 Therefore, this whole limit is equal to the absolute value of x minus 4.

02:23 And recall that by the ratio test, for the series to converge, we want this limit to be less than 1.

02:32 This gives us, first of all, this gives us the radius of convergence, which is going to be one...

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