Golden ratio series

Here are a couple elegant series involving the golden ratio I = (1 + a5)/2.

First, reciprocals of integer powers:

Then reciprocals of odd powers:

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

More posts related to the golden ratio:

• Imaginary gold
• Oscillating Fibonacci ratios
• Golden strings and the rabbit constant
• Power method and Fibonacci ratios
• Fibonacci numbers and numerical integration