$\sum a_n$ converges absolutely, does $\sum (a_n + \cdots + a_n^n)$ converge 
Suppose $\sum_{n=1}^\infty a_n$ converges absolutely. Does this imply that the series $$\sum_{n=1}^\infty (a_n + \cdots + a_n^n)$$ converges?

I believe the answer is yes, but I can't figure out how to prove it. Any help would be appreciated.
Thanks.

Another way to see this is to notice that each term is less than $a_n + a_n^2 + a_n^3 + ...$ which, when $a_n < 1$ is $\frac{a_n}{1-a_n}$. When $a_n < \frac{1}{2}$ then $\frac{a_n}{1-a_n} < 2a_n$.
