Integral Test For Convergence

The series

:

\sum_{n=1}^\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

	CATEGORIES ABOUT INTEGRAL TEST FOR CONVERGENCE

	integral calculus

	mathematical series

	convergence tests

	articles containing proofs

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Integral Test For Convergence