Editing Ratio test (section)

==== 4. Extended Bertrand's test ====

This extension probably appeared at the first time by Margaret Martin in 1941.<ref name="Mar1941">{{cite journal|url=https://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07477-X/S0002-9904-1941-07477-X.pdf |last1=Martin |first1=Margaret |date=1941 |title=A sequence of limit tests for the convergence of series |journal=Bulletin of the American Mathematical Society |volume=47 |issue=6|pages=452–457  |doi=10.1090/S0002-9904-1941-07477-X |doi-access=free }}</ref> A short proof based on Kummer's test and without technical assumptions (such as existence of the limits, for example) was provided by Vyacheslav Abramov in 2019.<ref name="Abr2008">{{cite journal|last1=Abramov |first1=Vyacheslav M. |date=May 2020 |title=Extension of the Bertrand–De Morgan test and its application |journal=The American Mathematical Monthly |volume=127 |issue=5 |pages=444–448 |doi=10.1080/00029890.2020.1722551 |arxiv=1901.05843 |s2cid=199552015 }}</ref>

Let <math>K\geq1</math> be an integer, and let <math>\ln_{(K)}(x)</math> denote the <math>K</math>th [[iteration|iterate]] of [[natural logarithm]], i.e. <math>\ln_{(1)}(x)=\ln (x)</math> and for any <math>2\leq k\leq K</math>, 
<math>\ln_{(k)}(x)=\ln_{(k-1)}(\ln (x))</math>.

Suppose that the ratio <math>a_n/a_{n+1}</math>, when <math>n</math> is large, can be presented in the form

:<math>\frac{a_n}{a_{n+1}}=1+\frac{1}{n}+\frac{1}{n}\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^i\ln_{(k)}(n)}+\frac{\rho_n}{n\prod_{k=1}^K\ln_{(k)}(n)}, \quad K\geq1.</math>
(The empty sum is assumed to be 0. With <math>K=1</math>, the test reduces to Bertrand's test.)

The value <math>\rho_{n}</math> can be presented explicitly in the form
:<math>\rho_{n} = n\prod_{k=1}^K\ln_{(k)}(n)\left(\frac{a_n}{a_{n+1}}-1\right)-\sum_{j=1}^K\prod_{k=1}^j\ln_{(K-k+1)}(n).</math>

Extended Bertrand's test asserts that the series
* Converge when there exists a <math>c>1</math> such that <math>\rho_n \geq c</math> for all <math>n>N</math>.
* Diverge when <math>\rho_n \leq 1</math> for all <math>n>N</math>.
* Otherwise, the test is inconclusive.

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

When the above limit does not exist, it may be possible to use limits superior and inferior. The series 
* Converge if <math>\liminf \rho_n > 1</math>
* Diverge if  <math>\limsup \rho_n < 1</math>
* Otherwise, the test is inconclusive.

For applications of Extended Bertrand's test see [[birth–death process]].