A geometric series is a series of the form arn (where we take a and r to be positive). One learns in high school algebra that this series converges if and only if 0 < r < 1. If arn does converge, we have
We can combine these comments about geometric series with the comparison test to yield another test called the ratio test: given a series an with an > 0 for all n, if there exists a number C with 0 < C < 1 such that
for all n, then an converges.
To prove this fact, note that under the hypotheses of the theorem,
|an≤Can-1≤C2an-2≤ ... ≤Cn-1a1|
Letting bn = a1Cn-1, so that bn is a (convergent) geometric series, we see that an≤bn. By the comparison test, an must also converge; in fact