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 V.P. Ermakov in [Er].

Let $f(x)$ be a positive decreasing function for $x \ge 1$. 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 if 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)