View source for Kummer criterion

{{MSC|40A05}}
{{TEX|done}}

A general convergence criterion for series with positive terms, proposed by E. Kummer. Let
\begin{equation}\label{e:series}
\sum_n a_n 
\end{equation}
be a series of positive numbers and $\{c_n\}$ a sequence of positive numbers. If there are $\delta >0$ and $N$ such that
\[
K_n := c_n \frac{a_n}{a_{n+1}} - c_{n+1} \geq \delta \qquad \forall n\geq N\, ,
\]
then \eqref{e:series} converges. If the series $\sum_n (c_n)^{-1}$ diverges and there is $N$ such that $K_n \leq 0$ for all $n\geq N$, then \eqref{e:series} diverges.

An obvious corollary is that, when the limit
\[
K := \lim_{n\to \infty} K_n
\]
exists we have:
* if $K>0$ \eqref{e:series} converges
* if $K<0$ and $\sum_n (c_n)^{-1}$ diverges, then \eqref{e:series} diverges.


====References====
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|valign="top"|{{Ref|Fi}}|| G.M. Fichtenholz,    "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag  Wissenschaft.  (1964)
|-
|valign="top"|{{Ref|Ra}}|| E.D. Rainville,   "Infinite series" , Macmillan  (1967)
|-
|}