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