   Chapter 11.6, Problem 6E

Chapter
Section
Textbook Problem

Determine whether the series is absolutely convergent or conditionally convergent. ∑ n = 1 ∞ ( − 1 ) n − 1 n n 2 + 4

To determine

Whether the given series is absolutely convergent or conditionally convergent.

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 if the series of absolute values Σ|an| is convergent.

b) Conditionally convergent series:

A series Σ an is called conditionally convergent if it is convergent.

4) Given:

n=1-1n-1nn2+4

5) Calculation:

Consider.

n=1-1n-1bn=n=1-1n-1nn2+4

Hence,bn=nn2+4

Also bn=nn2+4>0 for n1

For all n1,    n+1>n , i.e., n+12+4>n2+4,

n+1n+12+4<nn2+4

Therefore, bn is decreasing for n2 and

limnbn=limnnn2+4=limn1n+4/n=0

Hence, by the alternating series test,n=1-1n-1nn2+4 is convergent

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