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Integral test

A convergence test that compares a nonnegative decreasing series to an improper integral.
Integral test

Integral test: Let f:[1,∞)→[0,∞)f:[1,\infty)\to[0,\infty) be continuous and decreasing, and define an=f(n)a_n=f(n). Then the ∑n=1∞an\sum_{n=1}^\infty a_n if and only if the improper integral

∫1∞f(x) dx \int_1^\infty f(x)\,dx

converges (is finite).

Moreover, for each integer N≥1N\ge 1 one has the remainder bounds

∫N+1∞f(x) dx  ≤  ∑n=N+1∞an  ≤  ∫N∞f(x) dx. \int_{N+1}^\infty f(x)\,dx \;\le\; \sum_{n=N+1}^\infty a_n \;\le\; \int_N^\infty f(x)\,dx.

This test complements the for nonnegative series and is often paired with the for slowly varying terms.