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

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

Question
Series
asked 2021-02-02
Use the alternating series test to study the convergence of the following series
\(\sum_{n=1}^\infty(-1)^{n+1}ne^{-n}\)

Answers (1)

2021-02-03
Alternating series:
A series of the form
\(\sum_{n=1}^\infty(-1)^{n+1}na_n=a_0-a_1+a_2-a_3+...\)
where either all \(a_n\) are positive or all \(a_n\) are negative, is called an alternating series.
The alternating series test then says: if \(|a_n|\) decreases monotonically and \(\lim_{n\rightarrow\infty}a_n=0\) then the alternating series converges.
Moreover, let L denote the sum of the series, then the partial sum
\(S_k=\sum_{n=0}^k(-1)^na_n\)
approximates L with error bounded by the next omitted term:
\(|S_k-L|\leq|S_k-S_{k+1}|=a_{k+1}\)
i) \(a_n=ne^{-n}\)
\(=\frac{n}{e^n}>0\)
ii) \(n\to\infty\frac{n}{e^n}\)
\(n\to\infty\frac{1}{e^n}\)
\(=\frac{1}{\infty}\)
\(=0\)
iii) \(a_{n+1}
Hence the alternating series will converge.
0

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