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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