If $$\sum_{n=1}^{\infty}\frac{a_n}{e^n}$$ is convergent
denote$$S_n=\sum_{k=1}^{n}a_n$$
show that $$\sum_{n=1}^{\infty}\frac{S_n}{e^n}$$ is convergent.
If $$\sum_{n=1}^{\infty}\frac{a_n}{e^n}$$ is convergent
denote$$S_n=\sum_{k=1}^{n}a_n$$
show that $$\sum_{n=1}^{\infty}\frac{S_n}{e^n}$$ is convergent.
Use Abel's identity to show that:
$$\sum_{n=1}^{k}\frac{S_n}{e^n}=1/(e^{-1}-1)[e^{-(k+1)}S_{k+1}-e^{-1}S_1-\sum_{n=1}^{k}\frac{a_{n+1}}{e^{n+1}}]$$
Now the answer to your question follows very easily because the limit of the right hand side exists.