View source for Kummer criterion

A general convergence criterion for series with positive terms, proposed by E. Kummer. Given a 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/k/k055/k055950/k0559501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

and an arbitrary sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559502.png" /> of positive numbers such that the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559503.png" /> is divergent. If there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559504.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559505.png" />,

<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/k/k055/k055950/k0559506.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559507.png" /> is a constant positive number, then the series (*) is convergent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559508.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559509.png" />, the series (*) is divergent.

In terms of limits Kummer's criterion may be stated as follows. Let

<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/k/k055/k055950/k05595010.png" /></td> </tr></table>

then the series (*) is convergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k05595011.png" /> and divergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k05595012.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,   "Differential und Integralrechnung" , '''2''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.D. Rainville,   "Infinite series" , Macmillan  (1967)</TD></TR></table>