# Leibniz criterion

for convergence of an alternating series

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

#### References

 [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: http://www.encyclopediaofmath.org/index.php?title=Leibniz_criterion&oldid=29175
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article