Connection between series

Question

I have to show that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} a_n^2$ is absolutely convergent too.

Please give me some hint, how do I start the excercise. Thanks!

Answer 1

First show that there exist $n_0$ such that for all $n\geq n_0$ you have $|a_n|<1$. Then compare the tails of the two sums.

Answer 2

Here is a hint if the terms are all positive: suppose that $\displaystyle \sum_{n=1}^\infty |a_n|$ converges. Then $|a_n| \to 0$, and $\dfrac{a_n^2}{|a_n|} \to 0$ as well. Apply the limit comparison test.