View source for Leibniz criterion



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,   "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR></table>