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.

If you are reading this, your browser is not set to run Java applets. Try IE11 or Safari and declare the site https://www.cut-the-knot.org as trusted in the Java setup.

Sums of Geometric Series - Proofs Without Words

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

71543060