$\sum a_n$ converges absolutely, does $\sum (a_n + \cdots + a_n^n)$ converge

Question

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.

Answer 1

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.

Answer 2

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 $\frac{a_n}{1-a_n} < 2a_n$.