Answer
Diverges
Work Step by Step
Alternating series test:
Suppose that we have 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 condition are satisfied the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
Given: $\Sigma_{n=1}^{\infty}(-1)^{n-1}e^{2/n}$
$\lim\limits_{n \to \infty}b_{n}=\lim\limits_{n \to \infty}(-1)^{n-1}e^{2/n}=(-1)^{n-1}= DNE$
As $n$ increases , $2/n$ approaches to $0$ . Because of the alternating sign , the limit will oscillate between $-1$ to $1$.
The limit is not zero.
Hence, the given series diverges by the divergence test.
