Leibniz criterion

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2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

for convergence of an alternating series

If the terms of an alternating series \begin{equation} \sum_{n=1}^{\infty}(-1)^{n+1}a_n,\quad a_n>0, \end{equation} decrease monotonically ($a_n\geq a_{n+1}$, $n=1,2,\dots$) and tend to zero ( $\lim\limits_{n\to\infty}a_n=0$ ), then the series converges; moreover, a remainder of the series, \begin{equation} \sum_{k=n+1}^{\infty}(-1)^{k+1}a_k, \end{equation} has the sign of its first term and is less than it in absolute value. The criterion was established by G. Leibniz in 1682.



[Kn] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
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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