If one series is convergent then so is the other

Question

Prove/disprove:

Let $(a_n)_{n=1}^{\infty}$ be a non negative sequence and let $\sum_{n=1}^{\infty}{e^{-a_n}}$ be a converegent series.
Then the series $\sum_{n=1}^{\infty}\frac{1}{a_n^7}$ is convergent.

I was trying to prove by contradiction:
Assume $\sum_{n=1}^{\infty}\frac{1}{a_n^7}$ diverges. Then $\lim_{n\to \infty}a_n\neq \infty$. Therefore $\sum_{n=1}^{\infty}{e^{-a_n}} = \sum_{n=1}^{\infty}{\frac{1}{e^{a_n}}} = \infty$.

This feels really intuitive, but I feel like I'm lacking something.

Answer 1

The statement is false: if $a_n=\sqrt[7]n$, then $\displaystyle\sum_{n=1}^\infty e^{-a_n}$ converges, but $\displaystyle\sum_{n=1}^\infty\frac1{a_n^{\,7}}$ diverges (it's the harmonic series).