Answer
Convergent
Work Step by Step
Alternating series test:
Suppose that we have a series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$.
Then if the following two conditions are satisfied, the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
Given: $\Sigma_{n=1}^{\infty}(-1)^{n+1}ne^{-n}$
In the given problem, $b_{n}=ne^{-n}=\frac{ n}{e^{n}}$
which satisfies both conditions of Alternating Series Test as follows:
1. $b_{n}=\frac{ n}{e^{n}}$ is decreasing because the denominator is increasing.
2. $\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}\frac{ n}{e^{n}}$
Since the limit is in the form of $\frac{\infty}{\infty}$, we can use L-hospital's rule.
$=\lim\limits_{n \to \infty}\frac{ 1}{e^{n}}$
$=\frac{1}{\infty}$
$=0$
Hence, the given series is convergent by the Alternating Series Test.