$\sum a_n$ 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.

After a while, $|a_k| \lt \frac{1}{2}$.  
From that point on, 
$$|a_k+a_k^2+a_k^3+\cdots +a_k^k| \le |a_k|+\frac{1}{2}|a_k|+\frac{1}{4}|a_k|+\cdots+\frac{1}{2^{k-1}}|a_k|\le 2|a_k|.$$
So by comparison our series converges absolutely.
