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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

arn =    

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

≤C    

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

an≤bn =    

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