if $\sum_{i=1}^n a_i$ converges but $\sum_{i=1}^n (a_i)^2$ diverges, does that mean that $\sum_{i=1}^n a_i$ is conditionally

Question

I don't think this is true, but cannot find an example to disproof it. Either that or i need to prove that $\sum_{i=1}^n |a_i|$ diverges, which I am unclear how to approach the proof.

Answer 1

Yes, it is true. That's so because, if $\sum_{n=1}^\infty a_n$ was absolutely convergent, then we would have $\lim_{n\to\infty}a_n=0$ and therefore $a_n^{\,2}\leqslant |a_n|$ if $n$ is large enough. So, by the comparison test (this is where absolute convergence is used), $\sum_{n=1}^\infty a_n^{\,2}$ would converge too.