Sums and products involving Fibonacci In summary, if $\phi$ is the golden ratio, I want to show:
\begin{align}
\sum_{n=1}^\infty \frac1{F_n} &= 4-\phi \\
\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{F_nF_{n+1}} &= \phi-1 \\
\prod_{n=2}^\infty \bigg( 1 + \frac{(-1)^n}{F_n^2} \bigg) &= \phi.
\end{align}

The first equation is obviously wrong, since
$$\frac{1}{1} + \frac{1}{1} + \frac{1}{2} = \frac{5}{2} > 4 - \phi = 4 - \frac{1+\sqrt{5}}{2} \approx 2.381966.$$
The other two equations can be easily derived by considering
$$F_n^2 - F_{n-1}F_{n+1}.$$
