Suppose $\sum_{n=1}^{\infty}a_{n}$ converges, where $a_{n} \ge 0\ \forall\ n \in \mathbb{N} \ $ Prove that $\sum_{n=1}^{\infty}a_{n}^{p}$ converges for all $p 1$

Question

Suppose $\sum_{n=1}^{\infty}a_{n}$ converges, where $a_{n} \ge 0\  \forall\ n \in \mathbb{N} \ $ Prove that $\sum_{n=1}^{\infty}a_{n}^{p}$ converges for all $p > 1$

anonymous · Answer

I'm guessing I have to construct some sort of comparison, which means I would have to show $a_{n} \ge\ a_{n}^{p}$ in order to show $a_{n}^{p}$ converges. Just not sure how to do that or if my idea is even correct.

zarkon · Answer

if the sum converges then there exists an $N$ such that $a_n<1$ for all $n\ge N$

thus for $n\ge N$ we have $a_n^p<a_n$ when $p>1$

ganeshie8 · Answer

nice :)

anonymous · Answer

That claim might definitely be enough. I think I can use that and construct something, thanks ^_^

zarkon · Answer

use the fact that a necessary condition for $\sum_{n=1}^{\infty}a_{n}$ to converge is that $a_n\to 0$ as $n\to\infty$

anonymous · Answer

That's exactly where I was going with it, too! Thanks for the confirmation of that :)