lim n 
S
n
= 1/(1-r) if and only if | r | < 1.
Theorem X.B.1 (Convergence of the fundamental geometric series.) Suppose S n = 1 + r + r2 + r 3 + ... + r n . Then
lim n S
n
= 1/(1-r) if and only if | r | < 1.
In the figure below, consider the left most square to have side of length
1. By considering the similar triangles notice that AB/1 = 1/(1-r). But the sides of the squares accumulate to give AB = 1 + r
+ r2 + r 3 + ... .