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# 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}

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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)

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