TITLE: If $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty a_n^{n/({n+1})}$
QUESTION [6 upvotes]: This is the $1988$ Putnam $B4$ Problem:

Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty a_n^{n/({n+1})}$.

My problem lies in figuring out what to do in the case that $0\lt a_n \lt 1$ for all $n$. I imagine that it must have something to do with the limit comparison test. Any hints would be greatly appreciated.

REPLY [4 votes]: Limit comparison seems good. What can we say about $$\lim_{n\to\infty}\frac{a_n}{a_n^{n/(n+1)}}=\lim_{n\to\infty}a_n^{1/(n+1)}?$$