A geometric series is a series of the form ar n (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 ar n does converge, we have
|ar n =|
We can combine these comments about geometric series with the comparison test to yield another test called the ratio test: given a series a n with a n > 0 for all n , if there exists a number C with 0 < C < 1 such that
for all n , then a n converges.
To prove this fact, note that under the hypotheses of the theorem,
|a n≤Ca n-1≤C 2 a n-2≤ ... ≤C n-1 a 1|
Letting b n = a 1 C n-1 , so that b n is a (convergent) geometric series, we see that a n≤b n . By the comparison test, a n must also converge; in fact
|a n≤ b n =|