Integral test for convergence
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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) = an 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 0521588073de:Integralkriterium