   Chapter 11.3, Problem 36E

Chapter
Section
Textbook Problem

# (a) Find the partial sum s10 of the series ∑ n = 1 ∞ 1 / n 4 . Estimate the error in using s10 as an approximation to the sum of the series. (b) Use (3) with n = 10 to give an improved estimate of the sum. (c) Compare your estimate in part (b) with the exact value given in Exercise 35. (d) Find a value of n so that sn is within 0.00001 of the sum.

(a)

To determine

To find: The partial sum s10 of the series and estimate the error.

Explanation

Given:

The series is n=11n4 .

Result used:

(1) If the function f(k)=ak , where f is a continuous, positive and decreasing function for xn and an is convergent.

If Rn=ssn , then n+1f(x)dxRnnf(x)dx

(2) If the function is continuous, positive and decreasing function on [1,) and let ak=f(k) , then the series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

(3) The function is decreasing if f(x)<0 .

(4) The P series n=11np is convergent if p>1 .

Calculation:

Obtain the partial sum s10 .

s10=n=1101n4=114+124+134+...+1104=1+116+...+110000

Simplify further and obtain the required sum as shown below.

s10=1+124+134+144+154+164+174+184+194+1104=1+116+181+1256+1625+11296+12401+14096+16561+1100001

(b)

To determine

To estimate: The error of the sum of the series.

(c)

To determine

To compare: The estimated value obtained in part (b) with the exact value given in Exercise 35.

(d)

To determine

To find: The value of n so that sn is within 0.00001 of the sum.

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