   Chapter 11.6, Problem 31E

Chapter
Section
Textbook Problem

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

To determine

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

Explanation

1) Concept:

Use the alternating series test, definition of absolutely convergent series and conditionally convergent series.

2) Alternating series test:

If the alternating series

n=1-1n-1bn=b1-b2+b3-b4+b5-b6+         bn>0

satisfies

(i)  bn+1bn for all n

(ii) limnbn=0

then the series is convergent.

3) 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.

4) Given:

n=2-1nlnn

5) Calculation:

Consider,

n=2-1n-1bn=n=2-1nlnn

Hence, bn=1lnn

Also bn=1lnn>0 for n2

For all n2,    n+1>n , i.e

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