Ratio Test - The Test

The Test

The usual form of the test makes use of the limit

(1)

The ratio test states that:

• if L < 1 then the series converges absolutely;
• if L > 1 then the series does not converge;
• 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

and .

Then the ratio test states that:

• if R < 1, the series converges absolutely;
• if r > 1, the series diverges;
• if for all large n (regardless of the value of r ), the series also diverges; this is because is nonzero and increasing and hence does not approach zero;
• the test is otherwise inconclusive.

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