Editing Ratio test (section)

==== 3. Bertrand's test ====

This extension is due to [[Joseph Bertrand]] and [[Augustus De Morgan]].

Defining:

:<math>\rho_n \equiv n \ln n\left(\frac{a_n}{a_{n+1}}-1\right)-\ln n</math>

Bertrand's test<ref name="Bromwich1908"/><ref name="Duris2009"/> asserts that the series will:
* Converge when there exists a ''c>1'' such that <math>\rho_n \ge c</math> for all ''n>N''.
* Diverge when <math>\rho_n \le 1</math> for all ''n>N''.
* Otherwise, the test is inconclusive.

For the limit version, the series will:
* Converge if <math>\rho=\lim_{n\to\infty}\rho_n>1</math> (this includes the case ''ρ'' = ∞)
* Diverge if <math>\lim_{n\to\infty}\rho_n<1</math>.
* If ''ρ'' = 1, the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior.<ref name="Bromwich1908"/><ref name="Blackburn2012"/><ref>{{mathworld|title=Bertrand's Test|urlname=BertrandsTest}}</ref> The series will:
* Converge if <math>\liminf \rho_n > 1</math>
* Diverge if  <math>\limsup \rho_n < 1</math>
* Otherwise, the test is inconclusive.