Integral test

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

An elementary criterion for the convergence of the series $\sum_n f(n)$ when $f:[1, \infty[\to [0, \infty[$ is a monotone decreasing function: the series converges if and only if the integral \[ \int_1^\infty f(x)\, dx \] is finite.