Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
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Theorem
\(\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }\) | \(=\) | \(\ds \dfrac 1 {1 \times 3} + \dfrac 1 {3 \times 8} + \dfrac 1 {8 \times 21} + \dfrac 1 {21 \times 55} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{-2}\) |
where:
- $F_k$ denotes the $k$th Fibonacci number
- $\phi$ denotes the golden mean.
Proof
\(\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }\) | \(=\) | \(\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} } \paren {\dfrac {F_{2 k + 2} - F_{2 k} } {F_{2 k + 2} - F_{2 k} } }\) | multiplying by $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac 1 {F_{2 k + 2} } } \paren {\dfrac 1 {F_{2 k + 2} - F_{2 k} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac 1 {F_{2 k + 2} } } \paren {\dfrac 1 {\paren {F_{2 k + 1} + F_{2 k} } - F_{2 k} } }\) | Definition of Fibonacci Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac 1 {F_{2 k + 2} } } \paren {\dfrac 1 {F_{2 k + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} F_{2 k + 1} } - \dfrac 1 {F_{2 k + 1} F_{2 k + 2} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 2} \paren {-1}^k \dfrac 1 {F_k F_{k + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \phi^{-2}\) | Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared |
$\blacksquare$
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): $5$