[Math] $\sum a_{n}$ converges but $\sum a_{n}^2 $ diverges

Question

I have to give an example of a convergent series $\sum a_{n}$ for which $\sum a_{n}^2 $ diverges.

I think that such a series cannot exist because if $\sum a_{n}$ converges absolutely then $\sum a_{n}^2 $ will always converge right?

Answer 1

The alternating series test gives a wealth of examples. Take $$ a_n=(-1)^n/\sqrt{n} $$ for example.