**The Sum of A Geometric Series of Lengths.**
© 2000 M. Flashman

**Theorem X.B.1** **(Convergence of the fundamental geometric series.) **Suppose** ****S
**_{n}** **= **1 + ***r*
+ *r*^{2} + *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*
+ *r*^{2} + *r* ^{3} + ... .

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