p-Series

$\sum\dfrac{1}{n^p}$ — converges exactly when $p>1$ , diverges otherwise (including the harmonic series at $p=1$ ).

A p-series $\sum_{n=1}^{\infty}\dfrac{1}{n^p}$ converges if and only if $p>1$ , and diverges for $p\le1$ — provable directly via the Integral Test, since $\int_1^{\infty}x^{-p}\,dx$ converges exactly under the same condition. The harmonic series is the $p=1$ boundary case, diverging just barely.

Examples

$\sum\frac{1}{n^2}$ converges ( $p=2>1$ ); $\sum\frac{1}{\sqrt n}$ diverges ( $p=\frac12\le1$ ).