Millin Series

Theorem
The Millin series is the series defined as:


 * $\ds \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} } = \frac {7 - \sqrt 5} 2$

where $F_k$ is the $k$th Fibonacci number.

Proof
First we will prove that:


 * $\ds \sum_{r \mathop = 0}^n \frac 1 {F_{2^r} } = 3 - \frac {F_{2^n - 1} } {F_{2^n} }$

for $n \ge 1$.

We see that:


 * $\dfrac 1 {F_1} + \dfrac 1 {F_2} = 2 = 3 - \dfrac {F_1} {F_2}$

so the proposition holds for $n = 1$.

Suppose the proposition is true for $n = k$.

Then:

Thus by Principle of Mathematical Induction, the proof is complete.

Now taking the limit, we have:

{{eqn | r = \lim_{n \mathop \to \infty} \paren {3 - \frac {F_{2^n - 1} } {F_{2^n } } | c = }}

as required.