Find the radius of convergence and interval of convergence of the series. ∑n=1^∞ (x^n)/(n^4 4^n)

Name: Problem 9, Chapter 11: Calculus
Uploaded: 2021-07-20T19:22:58-08:00
Duration: 7 min 52 s
Channel: Tamara Worner
Description: Find the radius of convergence and interval of convergence of the series. ∑n=1^∞ (x^n)/(n^4 4^n)

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

Key Concepts

Ratio Test

The Ratio Test is a method used to determine the convergence of series by considering the limit of the absolute ratio of successive terms. For power series, it is particularly useful for finding the radius of convergence by assessing how the terms behave as n approaches infinity, thus indicating whether the series converges or diverges based on the value of |x - c|.

Interval of Convergence

The interval of convergence is the set of all x-values for which a power series converges. It is found by determining the radius of convergence and then testing the endpoints separately to see whether the series converges at those boundary points.

Radius of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges. It is determined using tests such as the Ratio Test or Root Test, and it provides a circle (or interval in the real case) inside which the power series converges absolutely.

Power Series

A power series is an infinite series of the form ? a? (x - c)? where a? are coefficients and c is the center of the series. It represents functions as sums of polynomial terms and is fundamental in analysis because it can converge to a function within some interval, depending on the values of x and the behavior of the coefficients.

Transcript

00:01 Okay, the key to this problem is finding an interval of convergence and a radius of convergence.

00:07 And to do that, we're going to follow this little step by step over here in the red box.

00:12 First, we need to determine when the limit of the n plus first term in the series divided by the nth term is less than one.

00:21 And then second, we have to check our end points individually.

00:25 So for this problem, we're wanting to find the interval of convergence for the the following series right here.

00:38 So let's take a look at that.

00:41 So again, first thing we're going to do is calculate the limit as n goes to infinity, absolute value of the n plus first term.

00:52 So we're going to replace all the ends by n plus 1.

00:55 So x to the n plus 1 over n plus 1 in parentheses to the 4 times 4 to the n plus 1.

01:14 And then all of that divided by x to the n over n to the fourth, four to the n.

01:24 Okay, so a lot of writing there, but a lot of it's going to cancel out.

01:29 Okay, so next step is always to simplify your fraction.

01:35 So we're going to go limit as n goes to infinity.

01:39 And i like to rewrite these like this, x to the n plus 1 over n plus 1 to the 4 to the n plus 1.

01:58 And then this fraction on the bottom here, this guy.

02:02 Remember we just flip and multiply to divide fractions.

02:06 So that's going to look like this, n to the fourth, 4 to the n over x to the n.

02:14 Now, lots of things are going to cancel out, right? and so this is how you kind of know you're doing it right.

02:20 So first of all, the 4 to the n and the 4 to the n plus 1 will cancel out and will be left with just a 4.

02:28 Because again remember i subtract my exponents when i'm dividing and then likewise same idea up here the x to the n plus one and the x to the end will cancel it and we'll be left with just an x to the first power so now let me rewrite that again limit as n goes to infinity i'm going to have an x over four and then i have these two things to the fourth power so i'm going to rewrite that like this n over n over n and then i have these two things to the fourth power so i'm going to rewrite that like this n over n.

03:00 And plus 1 to the fourth.

03:03 They were both to the fourth power, so let me just simplify it like that.

03:08 All right, the x over 4 doesn't have any ends in it, so it's constant, so it can come out front.

03:15 So that is going to look like the absolute value of x over the absolute value of 4, which is 4, times the limit as n goes to infinity of n over n plus 1 to the 4th, and this limit, you guys should recognize that limit is one...

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