Absolute convergence

A series converges absolutely when the series of absolute values converges; absolute convergence implies convergence.

If $a_n\ge 0$ and $\sum a_n<\infty$, then $a_n\to 0$, so eventually $a_n^2\le a_n$ and $\sum a_n^2$ converges.

The Cauchy-Schwarz inequality gives

If $a_n\ge 0$, $\sum a_n<\infty$, and $p>1/2$, thenThe endpoint can fail: $a_n=1/(n(\log n{)}^2)$ gives the divergent series $\sum 1/(n\log n)$.