Closed Form for Millin Series/Proof 2
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Theorem
The Millin series has the closed form expression:
- $\ds \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} } = \frac {7 - \sqrt 5} 2$
Proof
Let:
- $\ds \map F x := \sum_{k \mathop = 1}^\infty \frac {x^{2^{k - 1} } } {F_{2^k} }$
- $\alpha := \dfrac {1 + \sqrt 5} 2$
- $\beta := \dfrac {1 - \sqrt 5} 2$
Note that for $\size x \le 1$ we have:
- $\size {\dfrac {x^{2^{k - 1} } } {F_{2^k} } } \le \dfrac 1 {F_{2^k} }$
Given that the Millin Series converges, by Comparison Test:
- $\ds \map F x = \sum_{k \mathop = 1}^\infty \frac {x^{2^{k - 1} } } {F_{2^k} }$ converges for $\size x \le 1$
We have:
- $\ds \map F {\alpha x} := \sum_{k \mathop = 1}^\infty \frac {\alpha^{2^{k - 1} } x^{2^{k - 1} } } {F_{2^k} }$
Hence:
\(\ds \map F {\alpha x} + \map F {\beta x}\) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac {\alpha^{2^{k - 1} } + \beta^{2^{k - 1} } } {F_{2^k} } x^{2^{k - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac {L_{2^{k - 1} } } {F_{2^k} } x^{2^{k - 1} }\) | Closed Form for Lucas Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac {x^{2^{k - 1} } } {F_{2^{k - 1} } }\) | Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {x^{2^k} } {F_{2^k} }\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^{2^0} } {F_{2^0} } + \sum_{k \mathop = 1}^\infty \frac {\paren {x^2}^{2^{k - 1} } } {F_{2^k} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x + \map F {x^2}\) |
Since $\beta^2 = \dfrac {3 - \sqrt 5} 2 \le 1$, we can substitute $x = -\beta$:
\(\ds -\beta + \map F {\beta^2}\) | \(=\) | \(\ds \map F {-\beta^2} + \map F {-\alpha \beta}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map F 1\) | \(=\) | \(\ds -\beta + 2 \beta^2\) | as $\alpha \beta = -1$ and $\map F {\beta^2} = \map F {-\beta^2} + 2 \beta^2$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 0}^\infty \frac 1 {F_{2^k} }\) | \(=\) | \(\ds 1 + \map F 1\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \beta + 2 \beta^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 - \paren {1 + \beta - \beta^2} + \beta^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 + \dfrac {3 - \sqrt 5} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {7 - \sqrt 5} 2\) |
$\blacksquare$
Sources
- 1976: A.G. Shannon: Advanced Problems and Solutions: Sum Reciprocal! (Fibonacci Quart. Vol. 14, no. 2: pp. 186 – 187)