0
$\begingroup$

The question: Show that if $\sum_{n=1}^\infty a_n$ converges absolutely, then $\sum_{n=1}^\infty a_n^2$ converges absolutely.

I'm stuck, not sure what path to take in solving this. Can anyone provide a hint?

Thank you.

  • 0
    Thank you, but what if it's greater than 1?2012-11-13

1 Answers 1

2

HINT: Since $\sum_{n=1}^\infty a_n$ converges absolutely, $\lim_{n\to\infty}|a_n|=0$. Therefore $|a_n|<1$ for all sufficiently large $n$; why? Now use the fact that if $|a_n|<1$, then $a_n^2<|a_n|$.