Show that $\sum_{n=1}^\infty \frac{e^n + a*n}{e^{3n}}$ converges for all $a > 0$

Question

My thoughts: I can split the series into $\frac{1}{e^{2n}} + \frac{an}{e^{3n}}$, and I guess since $e^{3n}$ is exponentially bigger than $a*n$, it wouldn't matter what $a$ was. But with that reasoning, $a$ could be negative too.

Answer 1

Note that eventually $\forall a\in \mathbb{R}\quad e^n\ge an$ then for n sufficiently large the series is positive and we can apply, for example, limit comparison test with $\frac 1{n^2}$ to show convergence.