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

## 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: $z = 0$

$\blacksquare$