The Golden Ratio! 1

This sum is an infinite geometric series of the form $\displaystyle\sum_{n=1}^{\infty} x^{n}$ with $x = \dfrac{1}{\phi}$ .

Since $|x| \lt 1$ we know that this sum simplifies to

$\dfrac{x}{1 - x} = \dfrac{\dfrac{1}{\phi}}{1 - \dfrac{1}{\phi}} = \dfrac{1}{\phi - 1} = \dfrac{1}{\dfrac{1 + \sqrt{5}}{2} - 1} = \dfrac{2}{\sqrt{5} - 1} * \dfrac{\sqrt{5} + 1}{\sqrt{5} + 1} = \dfrac{\sqrt{5} + 1}{2} = \boxed{\phi}.$

(Note that $\phi^{2} - \phi - 1 = 0 \Longrightarrow \phi - 1 = \dfrac{1}{\phi} \Longrightarrow \dfrac{1}{\phi - 1} = \phi$ , so we could have skipped the last bit of algebra.)