Editing Ratio test (section)

== The test ==
[[File:Decision diagram for the ratio test.svg|thumb|Decision diagram for the ratio test]]
The usual form of the test makes use of the [[limit (mathematics)|limit]]
{{NumBlk|:|<math>L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.</math>|{{EquationRef|1}}}}
The ratio test states that:
* if ''L'' < 1 then the series [[absolute convergence|converges absolutely]]; 
* if ''L'' > 1 then the series [[divergent series|diverges]];
* if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if [[limit superior]] and [[limit inferior]] are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let

:<math>R = \lim\sup \left|\frac{a_{n+1}}{a_n}\right|</math>
:<math>r = \lim\inf \left|\frac{a_{n+1}}{a_n}\right|</math>.

Then the ratio test states that:<ref>{{harvnb|Rudin|1976|loc=§3.34}}</ref><ref>{{harvnb|Apostol|1974|loc=§8.14}}</ref>
* if ''R'' < 1, the series converges absolutely;
* if ''r'' > 1, the series diverges; or equivalently if <math>\left|\frac{a_{n+1}}{a_n}\right|> 1</math> for all large ''n'' (regardless of the value of ''r''), the series also diverges; this is because <math>|a_n|</math> is nonzero and increasing and hence {{mvar|a<sub>n</sub>}} does not approach zero;
* the test is otherwise inconclusive.

If the limit ''L'' in ({{EquationNote|1}}) exists, we must have ''L'' = ''R'' = ''r''. So the original ratio test is a weaker version of the refined one.