   Chapter 11.3, Problem 40E

Chapter
Section
Textbook Problem

# How many terms of the series ∑ n = 2 ∞ 1 / [ n ( ln   n ) 2 ] would you need to add to find its sum to within 0.01?

To determine

To find: The number of terms of the series required to add to obtain the sum within 0.01.

Explanation

Given:

The series n=11x(lnx)2 and remainder Rn0.01

Result used:

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

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

(3) 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 .

Calculation:

Consider the function from given series 1x(lnx)2=(x)1(lnx)2 .

The derivative of the function is obtained as follows:

f(x)=x1ddx((lnx)2)+(lnx)2ddx(x1)=x1(2(lnx)21(1x))+(lnx)2((1)x2)=(2x2(lnx)3+1x2lnx2)=(2+lnxx2(lnx)3)

Since f(x)<0 and by the Result (2), the given function is continuous, positive and decreasing on [1,)

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