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
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
arn = 
≤C
