Answer
The series diverges since it is the sum of a converging and diverging geometric series
Work Step by Step
We can split the series $\sum\limits^{\infty}_{n=1} \frac{2^n + 4^n}{e^n}$ into the sum of two geometric series $\sum\limits^{\infty}_{n=1} \frac{2^n}{e^n} +\frac{4^n}{e^n} $
We can rewrite the two series as $\sum\limits^{\infty}_{n=1} (\frac{2}{e})^n +(\frac{4}{e})^n $ and see that both are geometric series.
Recall that a geometric series converges if the common ratio $r$ satisfies $$|r| \lt 1$$
The first series has a common ratio $\frac{2}{e} \lt 1$ so the series converges. However, the second series has a common ratio $\frac{4}{e} \gt 1$ so the series diverges.
The sum of a converging series and a diverging series diverges, thus this series diverges.
