   Chapter 11.5, Problem 16E

Chapter
Section
Textbook Problem

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

To determine

To test:

The given series is convergent or divergent.

Explanation

1) Concept:

The Alternating Series Test:

If the alternating series

n=1-1nbn,  bn>0

satisfies,

i) bn+1<bn,  for all n

ii) limnbn=0

then the series is convergent.

2) Given:

n=1ncosnπ2n

3) Calculation:

It is given that,

n=1ncosnπ2n=n=1-1nn2n

Since cosnπ=-1n, the given series is of the form of alternating series.

The given series can be written as

n=1-1nn2n=n=1-1nbn

where,

bn=n2n

bn>0, for all n1

Now, to check bn  is decreasing series or not, by using the first derivative test, write bn in the form of the function of t

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