   Chapter 11.5, Problem 33E

Chapter
Section
Textbook Problem

For what values of p is each series convergent? ∑ n = 1 ∞ ( − 1 ) n n + p

To determine

To find:

The values of   p for which the given series is convergent.

Explanation

1) Concept:

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 otherwise the series is divergent.

2) Given:

n=1-1nn+p

3) Calculation:

p can be positive, negative, and zero. Let’s check the divergence of series for these possibilities.

If   p=0

The given series will be:

n=1-1nn=n=1-1nbn

where

bn=1n

Determine   bn+1

bn+1=1n+1

Comparing   bn and   bn+1

bn+1<bn

This verifies the first condition, which is, the given series is decreasing for   n1

For the second condition, look for the term   bn as  n.

limnbn=limn1n

=1

=0

Thus, both the conditions of the alternating series are satisfied

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