   Chapter 11.6, Problem 47E

Chapter
Section
Textbook Problem

# (a) Find the partial s5 sum of the series ∑ n − 1 ∞ 1 / ( n 2 n ) . Use Exercise 46 to estimate the error in using s5 as an approximation to the sum of the series. (b) Find a value of n so that sn is within 0.00005 of the sum. Use this value of n to approximate the sum of the series.

(a)

To determine

To find: The partial sum s5 of the series and estimate the error approximation to the sum of the series.

Explanation

Given:

The series is n=11n2n.

Result used:

(1) If the sequence {rn} is an increasing sequence and Remainder Rnan+11L, then limnrn=L<1.

(2) The function f(x) is increasing if f(x)>0.

Calculation:

The kth partial sum of the series sk=k=1n1k2k.

If n=5, then the partial sum s5 is computed as follows.

s5=k=`51k2k=12+12(2)2+13(2)3+14(2)4+15(2)5=0.5+0.125+0.041667+0.015625+0.006250.68854

That is, s50.68854.

Consider rn=an+1an where an=1n2n and an+1=1(n+1)2n+1

rn=1(n+1)2n+11n2n=n2n(n+1)2n+1=n2(n+1)

Obtain the limit of rn.

limnrn=limnn2(n+1)=limnn2n(1+1n)=12(1+1)=12

That is, limnrn=L=12

(b)

To determine

To find: The value of n so that the error of sn is within 0.00005 and approximate the sum of the series.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 