Find the radius of convergence and interval of convergence of the series. ∑n = 1^∞ (n)/(2^n(n^2

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?, where each term consists of a coefficient a? and a variable x raised to the nth power. Power series are important because they can represent functions within a certain interval of values (the interval of convergence) and are widely used in analysis and applied mathematics for approximations and solving 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 provides a precise boundary for the values of x for which the series converges and is typically determined by applying tests such as the ratio test or root test to the coefficients of the series.

Interval of Convergence

The interval of convergence is the set of all x values for which a power series converges, and it consists of the open interval determined by the radius of convergence plus a careful analysis of the series at the endpoints of that interval. Unlike the radius, which is a measure, the interval describes the actual x-values for convergence.

Ratio Test

The ratio test is a method used to determine the convergence or divergence of an infinite series by examining the limit of the absolute value of the ratio of successive terms. In the context of power series, the ratio test is often used to compute the radius of convergence by setting up a limit involving the coefficients and the variable x.

Endpoint Convergence Tests

After finding the radius of convergence using tests like the ratio test, determining convergence at the endpoints of the interval requires additional analysis. Endpoint convergence tests, such as the alternating series test or comparison test, help to assess whether the series converges, converges conditionally, or diverges when x equals the boundary values.

Transcript

00:01 The radius of convergence here is r, limit as n goes to infinity, of absolute value of a .n over a .n plus one, where a .n is n over 2 to the n times n squared plus one.

00:15 So this is limit as n goes to infinity of n over 2 to the n times n squared plus one.

00:24 And then dividing by a sub -in plus one is the same thing as multiplying by the reciprocal.

00:31 So multiply by 2 to the end.

00:33 Times n plus 1 quantity squared plus 1 and then dividing by n plus 1 so that's limit as n goes to infinity of 2 to the n plus 1 divided by 2 to the n is just 2 and then we can group this in and this n plus 1 together just to make it a little easier to look at and then we can we're left with n plus 1 squared plus 1 divided by n squared plus 1.

01:13 So up here this is a degree 2 polynomial with leading coefficient 1.

01:17 Down here, this is a degree 2 polynomial with leading coefficient 1.

01:20 So as n goes to infinity, this is going to go to 1.

01:24 And as in goes to infinity, n over n plus 1 is also going to go to 1.

01:28 So this turns into 2 times 1, which is just 2.

01:33 So that's the radius of convergence.

01:35 For the interval of convergence, we need to figure out whether or not we include x equals minus 2 and whether or not we include x equals minus 2, and whether or not we include x equals 2.

01:43 So if x is equal to minus 2, then our sum turns into sum from n equals 1 to infinity of n, and then we have minus 2 to the n divided by 2 to the n.

01:56 The 2 to the ns will cancel.

02:00 So then we'll just have this minus 1 to the n here, and this n squared plus 1 down here.

02:10 So now we're alternating sine if we just plug in x equals minus 2 here...

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