   Chapter 11.3, Problem 28E

Chapter
Section
Textbook Problem

# Explain why the Integral Test can’t be used to determine whether the series is convergent.28. ∑ n = 1 ∞ cos 2 n 1 + n 2

To determine

Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1cos2n1+n2 .

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. Otherwise, it is divergent.

(2) The function f(x)<0 then it is monotonically decreasing.

Calculation:

Consider the function from given series cos2x1+x2 .

The derivative of the function obtained as follows,

f(x)=(1+x2)ddx(cos2x)cos2xddx(1+x2)(1+x2)2=(1+x2)2cosxddx(cosx)cos2x(2x)(1+x2)2=[2((1+x2)cosxsinx+xcos2x)(1+x2)2]

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