Golden ratio series

Here are a couple elegant series involving the golden ratio φ = (1 + √5)/2.

First, reciprocals of integer powers:

\frac{1}{\phi} + \frac{1}{\phi^2} + \frac{1}{\phi^3} + \frac{1}{\phi^4} + \cdots = \phi

Then reciprocals of odd powers:

\frac{1}{\phi} + \frac{1}{\phi^3} + \frac{1}{\phi^5} + \frac{1}{\phi^7} + \cdots = 1

Both are easy to prove since they’re geometric series.

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One thought on “Golden ratio series

  1. It also then follows the sum of even powers of 1/phi (except for the zero power) is just 1/phi. Those three identities together are quite lovely.

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