/*! 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} Q. 34 In Exercises 31鈥�34 in Section ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q. 34 (page 692)

In Exercises 31鈥�34 in Section 8.2 you were asked to find the Maclaurin series for the specified function. Now find the Lagrange鈥檚 form for the remainder Rn(x), and show that limn鈫�鈭�Rn(x)=0on the specified interval.

ex,鈩�

Short Answer

Expert verified

We've proved thatlimn鈫�鈭�Rn(x)=0

Step by step solution

01

Given Information  

Given equation : ex,鈩�

Theory used : For n>0,if|f(n+1)(c)|鈮�1for every value of x then using the Lagrange's form for the remainder, we have

Rn(x)=f(n+1)(c)(n+1)!xn+1

02

Finding the Lagrange’s form for the remainder and proving limn→∞Rn(x)=0

We get the Lagrange form of remainder by :

Rn(x)=f(n+1)(c)(n+1)!(x-x0)n+1

Where, clies between xandx0

But,

f(x)=ex鈬�f(n+1)c=ec鈭赌n鈮�0

Also, since the series is Maclaurin's. So, :

Rn(x)=ec(n+1)!xn+1鈬�Rn(x)鈮�exxn+1(n+1)!

Taking the limit, we have :

limn鈫�鈭�Rn(x)鈮�exxn+1(n+1)!=0

as the quotientxn+1(n+1)!鈫�0whenn鈫�0

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

Find the interval of convergence for power series:鈭�k=0鈭�-1k2k+1!3x+72k+1

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0. Note: These are the same functions and values as in Exercises 41鈥�48.

51.sinx,蟺

Use an appropriate Maclaurin series to find the values of the series in Exercises 17鈥�22.

鈭�k=0鈭�(-1)kk1蟺k

If a function f has a Maclaurin series, what are the possibilities for the interval of convergence for that series?

Find the interval of convergence for power series:鈭�k=0鈭�k24k+33x+7k
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied 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.