/*! 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} Q5P Find the interval of convergence... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q5P (page 22)

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case鈭�n=1鈭�xn(n!)2

Short Answer

Expert verified

All power series convergence intervals are X values.

Step by step solution

01

Significance of ratio test

The ratio test as follows:

Solution steps:

First,find 蚁n=|an+1an|.

Second,蚁=limn鈫�鈭�pn. .

Third,p<1 t he series converges, ifp>1 the series diverges.

02

Finding convergence

Let us consider the power series

鈭�n=1鈭�xn(n!)2

And we would like to find the interval of convergence of the power series.

We will be using the ratio test, as follows:

Solution steps:

First,find role="math" localid="1658820727600" 蚁n=|an+1an|.

Second, 蚁=limn鈫�鈭�pn.

Third, p<1the series converges, if p>1the series diverges.

role="math" localid="1658820961551" 蚁n=|an+1an|=xn+1((n+1)!)2xn(n!)2蚁n=(xn+1(n!)2((n+1)!)2xn

03

Finding the points of divergence

Notice that(n+1)!=(n+1)n! . Hence

蚁n=x(n+1)2蚁=limn鈫�鈭�x(n+1)2=0

p<1For all the values of x .

04

Concluding statement

All power series convergence intervals are x values.

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

exsinx

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

limx鈫�0tanx-xx3.

esinx

鈭�n=1鈭�(-1)nxn(2n)!

By the method used to obtain (12.5)[which is the series(13.1)below], verify each of the other series (13.2)to (13.5)below.
See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Work Energy and Power

Read Explanation

Magnetism

Read Explanation

Kinematics Physics

Read Explanation

Energy Physics

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.