/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Determine the radius of converge... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 21

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{k^{2} x^{2 k}}{k !}$$

Short Answer

Expert verified
Answer: The radius of convergence for the given power series is 鈭�, and the interval of convergence is (-鈭�, 鈭�).

Step by step solution

01

Apply the Ratio Test

Recall the Ratio Test: If the limit as k goes to infinity of the absolute value of the ratio of successive terms is less than 1, then the series converges. First, let's write down the general term of the given power series: $$a_k = \frac{k^2 x^{2k}}{k!}.$$ Next, we calculate the ratio of successive terms: $$\frac{a_{k+1}}{a_k} = \frac{\frac{(k+1)^2 x^{2(k+1)}}{(k+1)!}}{\frac{k^2 x^{2k}}{k!}}.$$
02

Simplify the Ratio

Now, let's simplify the ratio of successive terms: $$\frac{a_{k+1}}{a_k} = \frac{(k+1)^2 x^{2k+2}}{(k+1)!} \cdot \frac{k!}{k^2 x^{2k}} = \frac{(k+1)^2}{(k+1)(k+1)} \cdot \frac{x^2}{k+1} = \frac{x^2}{k+1}.$$
03

Calculate the Limit

Next, we need to calculate the limit as k goes to infinity of the absolute value of the ratio of successive terms: $$\lim_{k \to \infty} \left|\frac{x^2}{k+1}\right|.$$ This limit is equal to 0 for all x because the denominator goes to infinity.
04

Determine the Radius of Convergence

Since the limit is less than 1 for all x, the power series converges for all x. This implies that the radius of convergence is infinity, i.e., R = 鈭�.
05

Test the Endpoints

Since the radius of convergence is infinite, there are no endpoints to test. Thus, the interval of convergence is the whole real line, represented as (-鈭�, 鈭�). So, the radius of convergence for the given power series is 鈭�, and the interval of convergence is (-鈭�, 鈭�).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan ^{-1} x \approx x$$

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$(1+4 x)^{-2}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{1}{x^{4}+2 x^{2}+1}$$

Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{x^{k}}{k(k-1)}$$

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{0}^{0.2} \frac{\ln (1+t)}{t} d t$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.