Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
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Theorem
\(\ds \sum_{k \mathop \ge 2} \left({-1}\right)^k \dfrac 1 {F_k F_{k + 1} }\) | \(=\) | \(\ds \dfrac 1 {1 \times 2} - \dfrac 1 {2 \times 3} + \dfrac 1 {3 \times 5} - \dfrac 1 {5 \times 8} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{-2}\) |
where:
- $F_k$ denotes the $k$th Fibonacci number
- $\phi$ denotes the golden mean.
Proof
\(\ds \sum_{k \mathop \ge 2} \paren {-1}^k \dfrac 1 {F_k F_{k + 1} }\) | \(=\) | \(\ds \sum_{k \mathop \ge 2} \paren {F_{k + 1} F_{k - 1} - F_k^2} \dfrac 1 {F_k F_{k + 1} }\) | Cassini's Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 2} \paren {\dfrac {F_{k - 1} } {F_k} - \dfrac {F_k} {F_{k + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {F_1} {F_2} - \lim_{k \mathop \to \infty} \dfrac {F_k} {F_{k + 1} }\) | Telescoping Series | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \dfrac 1 \phi\) | Ratio of Consecutive Fibonacci Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{-1} \paren {\phi - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{-2}\) | Definition of Golden Mean |
$\blacksquare$
Historical Note
This result is attributed to Pincus Schub.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$