Go to Geometric Series and Series for a general discussion.

Convergence of Geometric Sequences

A geometric series will converge if the common ration r is a proper fraction.  In other words, |r| < 1 or -1 < r < 1.

The sum of n terms of a series is given by S = t1(1 - rn)
                                                                         1 - r

If |r| < 1, then rn -> 0 as n -> oo

and S  ->    t1
                1 - r

So the series S will converge to   t1
                                                1 - r

Problems