Integral test for convergence

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The integral test for convergence is a method used to test infinite series of non-negative terms for convergence. An early form of the test of convergence was developed in India by Madhava in the 14th century, and by his followers at the Kerala School. In Europe, it was later developed, possibly independently, by Maclaurin and Cauchy and is sometimes known as the Maclaurin-Cauchy test.

The series

[\sum_^\infty a_n]

converges if and only if the integral

[\int_1^\infty f(x)\,dx]

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

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