# Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared

## 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.