Integral test for convergence

The integral test for convergence is a method used to test infinite series of nonnegative terms for convergence. It was developed independently by Maclaurin and Cauchy and is sometimes known as the Maclaurin-Cauchy test.

The series

<math>\sum_{n=1}^\infty a_n<math>

converges if and only if the integral

<math>\int_1^\infty f(x)\,dx<math>

is finite, where f(x) is a positive monotone decreasing function defined on the interval [1, ∞) and f(n) = a_n for all n. If the integral diverges, then the series will diverge as well.

References

• Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.3) ISBN 0486601536

• Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 4.43) ISBN 0521588073