Answer
Converges
Work Step by Step
We know that $\cos n\leq 1$ for all $n\geq 1$.
Then,
$\frac{1+\cos n}{e^n}\leq\frac{1+1}{e^n}$
so
$\frac{1+\cos n}{e^n}\leq \frac{2}{e^n}$
The series $\sum_{n=1}^\infty\frac{2}{e^n}=\sum_{n=1}^\infty 2\cdot (\frac{1}{e})^n$ is an infinite geometric series and it converges because the common ratio is $0<\frac{1}{e}<1$.
