# Leibniz criterion

From Encyclopedia of Mathematics

*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 ($a_n>a_{n+1}$, $n=1,2,\dots$) 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.

#### Examples

- Leibniz series $\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}$.

#### 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