Bertrand's Test

A Convergence Test also called de Morgan's and Bertrand's Test. If the ratio of terms of a Series $\{a_n\}_{n=1}^\infty$ can be written in the form

$${a_n\over a_{n+1}}=1+{1\over n}+{\rho_n\over n\ln n},$$

then the series converges if $\underline{\lim_{n\to\infty}} \rho_n>1$ and diverges if $\overline{\lim_{n\to\infty}} \rho_n<1$, where $\underline{\lim_{n\to\infty}}$ is the Lower Limit and $\overline{\lim_{n\to\infty}}$ is the Upper Limit.

See also Kummer's Test

References

Bromwich, T. J. I'a and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40, 1991.