Use the alternating series test to study the convergence of the following series sum_{n=1}^infty(-1)^{n+1}ne^{-n}

Marvin Mccormick 2021-02-02 Answered
Use the alternating series test to study the convergence of the following series
n=1(1)n+1nen
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escumantsu
Answered 2021-02-03 Author has 98 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
i) an=nen
=nen>0
ii) nnen
n1en
=1
=0
iii)  an+1

Hence the alternating series will converge.

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Jeffrey Jordon
Answered 2021-12-27 Author has 2064 answers

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