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} = |

Take a Study Break!