Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ Where $\theta(z,\tau)$ is the Jacobi theta function.
An interesting fact about this series is that $\sum_{n=0}^\infty \frac{1}{F_{2n+1}}$ is transcendental and $\sum_{n=0}^\infty \frac{1}{F_{2n+1}+1}$ is algebraic.