Kummer criterion

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

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

[Fi]	G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
[Ra]	E.D. Rainville, "Infinite series" , Macmillan (1967)