Ermakoff's Test

The series $\sum f(n)$ for a monotonic nonincreasing $f(x)$ is convergent if

$$\overline{\lim_{x\to\infty}}{e^x f(e^x)\over f(x)}<1$$

and divergent if

$$\underline{\lim_{x\to\infty}}{e^x f(e^x)\over f(x)}>1.$$

References

Bromwich, T. J. I'a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 43, 1991.