Direct Comparison Test

Bound an unfamiliar series above or below by a known one to determine convergence without a formula.

The Direct Comparison Test determines a series' convergence by comparing it, term by term, to a series already known to converge or diverge: if $0\le a_n\le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges too; if $0\le b_n\le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges too. It works only when a suitable, simpler bounding series can be found.

Examples

$\sum\frac{1}{n^2+1}$ converges by comparison, since $\frac{1}{n^2+1}\le\frac{1}{n^2}$ and $\sum\frac{1}{n^2}$ (a p-series with $p=2$ ) converges.