# Sums of Geometric Series - Proofs Without Words

The applet below illustrates three identities - sums of geometric series - with factors 1/2, 1/3, and 1/4, namely,

2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + ... = 1,

3^{-1} + 3^{-2} + 3^{-3} + 3^{-4} + ... = 1/2,

4^{-1} + 4^{-2} + 4^{-3} + 4^{-4} + ... = 1/3.

What if applet does not run? |

**Credits**: The proofs for factors 1/2 (Warren Page) and 1/4 (Sunday A. Ajose) have been published previously; both have been included into R. B. Nelsen's *Proofs Without Words*.

### Reference

- R. B. Nelsen,
*Proofs Without Words*, MAA, 1993

### Proofs Without Words

- Proofs Without Words
- Sums of Geometric Series - Proofs Without Words
- Sine of the Sum Formula
- Parallelogram Law: A PWW
- Parallelogram Law
- Ceva's Theorem: Proof Without Words
- Viviani's Theorem
- A Property of Rhombi
- Triangular Numbers in a Square
- PWW: How Geometry Helps Algebra
- Varignon's Theorem, Proof Without Words

|Activities| |Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny

71526427