Closed Form for Millin Series/Proof 1

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:

as required.