Ermakov convergence criterion
2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]
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 be a positive decreasing function for
. If the inequality V.P. Ermakov in [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.
References
[Br] | T.J. Bromwich, "An introduction to the theory of infinite series" , Macmillan (1947) |
[Er] | V.P. Ermakov, "A new criterion for convergence and divergence of infinite series of constant sign" , Kiev (1872) (In Russian) |
Ermakov convergence criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ermakov_convergence_criterion&oldid=35853