/*! 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 5 Do the interval and radius of co... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 5

Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.

Short Answer

Expert verified
In summary, the radius of convergence remains the same when a power series is differentiated or integrated. However, the interval of convergence may change at the endpoints, as differentiating or integrating can affect the convergence of the series at those points.

Step by step solution

01

Write the general power series

Consider a general power series in x: $$ \sum_{n=0}^{\infty} a_n (x - c)^n $$ where \(a_n\) are constant coefficients, and \(c\) is the center of the series.
02

Find the radius of convergence using the ratio test

We use the ratio test for convergence to determine the radius of convergence \(R\): $$ R = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| $$
03

Differentiate the power series term-by-term

Now we differentiate the power series term-by-term: $$ \frac{d}{dx}\left( \sum_{n=0}^{\infty} a_n (x - c)^n \right) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1} $$
04

Find the radius of convergence of the differentiated series using the ratio test

Now let's apply the ratio test to the differentiated series: $$ R' = \lim_{n\to\infty} \left|\frac{n a_n}{(n+1) a_{n+1}}\right| = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| $$ As we can see, the radius of convergence remains the same after differentiation, as \(R' = R\).
05

Integrate the power series term-by-term

Now we integrate the power series term-by-term: $$ \int \left(\sum_{n=0}^{\infty} a_n (x - c)^n \right) dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x - c)^{n+1} + C $$
06

Find the radius of convergence of the integrated series using the ratio test

Now let's apply the ratio test to the integrated series: $$ R'' = \lim_{n\to\infty} \left|\frac{(n+1) a_n}{(n+2) a_{n+1}}\right| = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| $$ As we can see, the radius of convergence remains the same after integration, as \(R'' = R\).
07

Answer and Explanation

The radius of convergence does not change when a power series is differentiated or integrated. The interval of convergence may change, however, at the endpoints of the interval, since differentiating or integrating can affect the convergence of the series at those endpoints.

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

Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series. $$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots, \text { for }-1 < x \leq 1$$ $$\sqrt{4-16 x^{2}}$$

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. $$f(x)=\cos 2 x+2 \sin x$$

Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty}(\sqrt{x}-2)^{k}$$

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. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\frac{1}{(3+4 x)^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.