Leibniz criterion

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for convergence of an alternating series

If the terms of an alternating series

decrease monotonically (, ) and tend to zero (), then the series converges; moreover, a remainder of the series,

has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682.



[a1] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Leibniz criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article