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.

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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

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