Answer
Converges by the Theorem 9.14
Work Step by Step
Rewrite summation, $\Sigma^{\infty}_{n=1} (-1)^n \frac{1}{e^n}$
Apply Alternating Series Test
i.) $\lim\limits_{n \to \infty} \frac{1}{e^n} = 0$
ii.) $a_{n+1} = \frac{1}{e^{n+1}} \leq \frac{1}{e^n} = a_n$
The series satisfies the Alternating Series Test, so it converges
