$\sum a_{n}$ is convergent, $\sum a_{n}^2$ is divergent. Prove $\sum a_{n}$ is conditionally convergent.

Question

Suppose that the series $\sum_{n=1}^{\infty}a_{n}$ converges while $\sum_{n=1}^{\infty}a_{n}^2$ diverges. Prove that $\sum_{n=1}^{\infty}a_{n}$ converges conditionally.

Answer 1

If $a_n$ converges absolutely, we have $\vert a_n \vert < 1$ eventually. This implies $a_n^2 < \vert a_n \vert$ eventually (for say $n>N)$, which in turn implies that $\sum_{n>N}a_n^2 < \sum_{n>N} \vert a_n \vert$, which implies $\sum_n a_n^2$ converges.