View source for Leibniz criterion

''for convergence of an alternating series''

If the terms of an alternating series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580901.png" /></td> </tr></table>

decrease monotonically (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580902.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580903.png" />) and tend to zero (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580904.png" />), then the series converges; moreover, a remainder of the series,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058090/l0580905.png" /></td> </tr></table>

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



====Comments====


====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>