Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

Consider an integer N and a non-negative, continuous function f defined on the unbounded interval [N, ∞), on which it is monotone decreasing. Then the infinite series

converges to a real number if and only if the improper integral

is finite. In other words, if the integral diverges, then the series diverges as well.

If the improper integral is finite, then the proof also gives the lower and upper bounds

${\displaystyle \int _{N}^{\infty }f(x)\,dx\leq \sum _{n=N}^{\infty }f(n)\leq f(N)+\int _{N}^{\infty }f(x)\,dx}$

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