Continued Fraction Expansion of Golden Mean/Successive Convergents

Theorem
Consider the continued fraction expansion to the golden mean:
 * $\phi = \left[{1, 1, 1, 1, \ldots}\right] = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots}}}$

The $n$th convergent is given by:
 * $C_n = \dfrac {F_{n + 1} } {F_n}$

where $F_n$ denotes the $n$th Fibonacci number.