Comparison test for series

For nonnegative sequences with $0\le a_n\le b_n$, convergence of ${\sum }_n{b}_n$ implies convergence of ${\sum }_n{a}_n$, while divergence of ${\sum }_n{a}_n$ implies divergence of ${\sum }_n{b}_n$.

If $f:[1,\infty )\to [0,\infty )$ is decreasing, then ${\sum }_{n\ge 1}f(n)$ and ${\int }_1^{\infty }f(x)dx$ either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.