Geometric Series

A series is called geometric if each term in the series is obtained from the preceding one by multiplying it by a common ratio. For example, the series

1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

is geometric, since each term is obtained by multiplying the preceding term by 1/2. In general, a geometric series is of the form

a + ar + ar^2 + ar^3 + ... = the sum over n from 1 to infinity of ar^(n-1) for a not equal to 0.

Geometric series are useful because of the following result:

The geometric series

the sum over n from 1 to infinity of ar^(n-1) = a + ar + ar^2 + ar^3 + ...

is convergent if |r| < 1, and its sum is

the sum over n from 1 to infinity of ar^(n-1) = a/(1-r) for |r| < 1

Otherwise, the geometric series is divergent.

So, for our example above, a=1, and r=1/2, and the sum of the series is

1 + 1/2 + 1/4 + ... = the sum over n from 1 to infinity of 1*(1/2)^(n-1) = 1/(1-1/2) = 1/(1/2) = 2.

Copyright © 1996 Department of Mathematics, Oregon State University

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