How do you do the Alternating Series Test on this series and what is the result? sum_{n=2}^inftyfrac{(-1)^n}{n+1}

Lipossig 2021-02-15 Answered
How do you do the Alternating Series Test on this series and what is the result?
n=2(1)nn+1
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Expert Answer

2k1enyvp
Answered 2021-02-16 Author has 94 answers

Alternating series:
A series of the form
n=1(1)n+1nan=a0a1+a2a3+...
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test then says: if |an| decreases monotonically and limnan=0 then the alternating series converges.
Moreover, let L denote the sum of the series, then the partial sum
Sk=n=0k(1)nan
approximates L with error bounded by the next omitted term:
|SkL||SkSk+1|=ak+1
an=1n+1
an+1=1n+2

Therefore an+1<an

So an is monotonically decrea sing

an>0 for all n=1,2,3,...

n

=0

So the series converges.

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Jeffrey Jordon
Answered 2021-12-27 Author has 2047 answers

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