Editing Ratio test (section)

=== Ali--Deutsche Cohen φ-ratio test ===

This test is an extension of the <math>m</math>th ratio test.<ref>{{cite journal|url=https://www.ems-ph.org/journals/show_abstract.php?issn=0013-6018&vol=67&iss=4&rank=2|last1=Ali |first1=Sayel |last2=Cohen |first2=Marion Deutsche |date=2012 |title=phi-ratio tests |journal=[[Elemente der Mathematik]] |volume=67 |issue= 4|pages=164–168 |doi=10.4171/EM/206 |doi-access=free }}</ref>

Assume that the sequence <math>a_n</math> is a positive decreasing sequence.

Let <math>\varphi:\mathbb{Z}^+\to\mathbb{Z}^+</math> be such that <math>\lim_{n\to\infty}\frac{n}{\varphi(n)}</math> exists. Denote <math>\alpha=\lim_{n\to\infty}\frac{n}{\varphi(n)}</math>, and assume <math>0<\alpha<1</math>.

Assume also that <math>\lim_{n\to\infty}\frac{a_{\varphi(n)}}{a_n}=L.</math>

Then the series will:

* Converge if <math>L<\alpha</math>
* Diverge if <math>L>\alpha</math>
* If <math>L=\alpha</math>, then the test is inconclusive.