Sum of Reciprocals of Sequence of Pairs of Odd Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean

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Theorem

\(\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} }\) \(=\) \(\ds \dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 5} + \dfrac 1 {5 \times 13} + \dfrac 1 {13 \times 34} + \cdots\)
\(\ds \) \(=\) \(\ds \phi^{-1}\)

where:

$F_k$ denotes the $k$th Fibonacci number
$\phi$ denotes the golden mean.


Proof

\(\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} }\) \(=\) \(\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} } \paren {\dfrac {F_{2 k + 1} - F_{2 k - 1} } {F_{2 k + 1} - F_{2 k - 1} } }\) multiplying by $1$
\(\ds \) \(=\) \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k - 1} } - \dfrac 1 {F_{2 k + 1} } } \paren {\dfrac 1 {F_{2 k + 1} - F_{2 k - 1} } }\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k - 1} } - \dfrac 1 {F_{2 k + 1} } } \paren {\dfrac 1 {\paren {F_{2 k} + F_{2 k - 1} } - F_{2 k - 1} } }\) Definition of Fibonacci Number
\(\ds \) \(=\) \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k - 1} } - \dfrac 1 {F_{2 k + 1} } } \paren {\dfrac 1 {F_{2 k} } }\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k - 1} F_{2 k} } - \dfrac 1 {F_{2 k} F_{2 k + 1} } }\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop \ge 1} \paren {-1}^{k + 1} \dfrac 1 {F_k F_{k + 1} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {1 \times 1} - \sum_{k \mathop \ge 2} \paren {-1}^k \dfrac 1 {F_k F_{k + 1} }\) Definition of Fibonacci Number
\(\ds \) \(=\) \(\ds 1 - \phi^{-2}\) Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared
\(\ds \) \(=\) \(\ds \phi^{-1}\) Power of Golden Mean as Sum of Smaller Powers: $z = 0$

$\blacksquare$


Also see

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