T/F : If $a_{n} \geq 0$, $p > 1$ and $\sum_{n=1}^{\infty}a_{n}$ converges, then $\sum_{n=1}^{\infty}(a_{n})^{p}$ converges

Question

T/F : If $a_{n} \geq 0$, $p > 1$ and $\sum_{n=1}^{\infty}a_{n}$ converges, then $\sum_{n=1}^{\infty}(a_{n})^{p}$ converges

In every example I'm trying to make it ends up being true, but I have a feeling that it is false. I need some directions/hints

Answer 1

Here is a hint: if each $a_n \le 1$, then $a_n^p \le a_n$ for all $n$.