   Chapter 11.5, Problem 16E

Chapter
Section
Textbook Problem

# Test the series for convergence or divergence.16. ∑ n = 1 ∞ n cos n π 2 n

To determine

To test: Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1ncosnπ2n .

Result used:

(1) “If the alternating series n=1(1)n1bn=b1b2+b3b4+...   bn>0 satisfies the conditions bn+1bn   for all n and limnbn=0 , then the series is convergent; otherwise, the series is divergent.”

(2) The function f(x) decreases when f(x)<0 .

Calculation:

Consider the given series n=1ncosnπ2n and an=ncosnπ2n .

an=ncosnπ2n=(1)nn2n     [cosnπ=(1)n]=(1)nn2n

Therefore, an=(1)nn2n and bn=n2n>0 .

Consider the function from given series, f(x)=x2x=x2x .

The derivative of the function as follows,

f(x)=ddx(x2x)=xddx(2x)+2xddx(x)   (by Product Rule)=xddx(exln2)+2x(1)         [ab=eblna]

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