Theorem 4.2.9: Geometric Series
Let a be any real number. Then the series
is called Geometric Series.
 if  a  < 1 the geometric series converges
 if  a  1 the geometric series diverges
=
Note that the index for the geometric series starts at 0. This is not important for the convergence behavior, but it is important for the resulting limit.
Examples 4.2.10:  

The proof consists of a nice trick. Consider the partial sum S _{N} and multiply it by a:
 S _{N} = 1 + a + a ^{2} + a ^{3} + ... + a ^{N}
 a S _{N} = a + a ^{2} + a ^{3} + ... + a ^{N+1}