Editing Ratio test (section)

=== Frink's ratio test ===
Another ratio test that can be set in the framework of Kummer's theorem was presented by [[Orrin Frink]]<ref name="Frink1948">{{cite journal |last1=Frink |first1=Orrin |date=October 1948|title=A ratio test |journal=Bulletin of the American Mathematical Society |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-54/issue-10/A-ratio-test/bams/1183512386.full |volume=54 |issue=10 |pages=953-953}}</ref> 1948.

Suppose <math>a_n</math> is a sequence in <math>\mathbb{C}\setminus\{0\}</math>,
* If <math> \limsup_{n\rightarrow\infty}\Big(\frac{|a_{n+1}|}{|a_n|}\Big)^n<\frac1e </math>, then the series <math>\sum_na_n</math> converges absolutely.
* If there is <math>N\in\mathbb{N}</math> such that <math> \Big(\frac{|a_{n+1}|}{|a_n|}\Big)^n\geq\frac1e </math> for all <math>n\geq N</math>, then <math>\sum_n|a_n|</math> diverges.

This result reduces to a comparison of <math>\sum_n|a_n|</math> with a [[power series]] <math>\sum_n n^{-p}</math>, and can be seen to be related to Raabe's test.<ref name="Stark1949">{{cite journal |date=1949|last1=Stark |first1=Marceli |title=On the ratio test of Frink |journal=Colloquium Mathematicum | volume=2 |issue=1 |pages=46-47 }}</ref>