View source for Ermakov convergence criterion

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

A criterion for the convergence of a series $\sum_n f(n)$, where $f:[1, \infty[\to [0, \infty[$ is a monotone decreasing function, established by 

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036200/e0362001.png" /> be a positive decreasing function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036200/e0362002.png" />. If the inequality V.P. Ermakov in {{Cite|Er}}. If there is $\lambda< 1$ such that
\[
\frac{e^x f(e^x)}{f(x)} < \lambda
\]
for sufficiently large $x$, then the series $\sum_n f(n)$ converges. If instead
\[
\frac{e^x f(e^x)}{f(x)}\geq 1
\]
for all sufficiently large $x$, then the series diverges. In particular the convergence or divergence of the series can be decided of the limit
\[
\lim_{x\to\infty} \frac{e^x f(e^x)}{f(x)}
\]
exists and differs from 1.

Ermakov's criterion can be derived from the [[Integral test|integral test]]. 

====References====
{|
|-
|valign="top"|{{Ref|Br}}|| T.J. Bromwich,   "An introduction to the theory of infinite series" , Macmillan  (1947)
|-
|valign="top"|{{Ref|Er}}|| V.P. Ermakov,   "A new criterion for convergence and divergence of infinite series of constant sign" , Kiev  (1872)  (In Russian)
|-
|}