   Chapter 11.6, Problem 37E

Chapter
Section
Textbook Problem

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. ∑ n = 1 ∞ ( − 1 ) n arctan n n 2

To determine

Whether the given series is absolutely convergent, conditionally convergent or divergent.

Explanation

1) Concept:

Use the definition of absolutely convergent series and conditionally convergent series.

2) Definition:

a) Absolutely convergent series:

A series Σ an is called absolutely convergent when the series of absolute values Σ|an| is convergent.

b) Conditionally convergent series:

A series Σ an is called conditionally convergent when it is convergent but not  absolutely convergent.

3) Given:

n=1-1narctannn2

4) Calculation:

Consider,

n=1bn=n=1-1narctannn2

Hence,bn=-1narctannn2

0<-1narctannn2<π/2n2 for n1

n=1

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