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
≤C |
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} = |