Relationship between sums involving reciprocal of fibonacci numbers

Question

I'm asked to prove that if $x_n$ denotes the fibonacci sequence and $$s = \sum_{n=1}^{+\infty}\frac1{x_{2n-1}^2}\text{, } s' = \sum_{n=1}^{+\infty}\frac{(-1)^{n-1}n}{x_{2n}}$$ then $s/s' = \sqrt{5}$. I don't think there is a closed expression for each of these sums, but I struggle to figure out this relationship, which using Binet's formula boils down to prove that:$$\sum_{n=1}^{+\infty}\left[\frac{\phi^{2n-1}}{1+(\phi^2)^{2n-1}}\right]^{2}
=\sum_{n=1}^{+\infty}\frac{(-1)^{n-1}n\cdot\phi^{2n}}{(\phi^2)^{2n}-1}$$
Any hints?

Answer 1

Hint: You can simplify the first sum by using $$\frac{1}{1+x}=\sum_{k=0}^\infty (-x)^k,$$ if $|x|<1$ (try to make the denominator $1+(\phi^2)^{-(2n-1)}$ instead). When you expand all this out, you should get terms that look like what you get when you do something similar with the second sum.