Infinite Fibonacci sums $\sum_{n=1}^{\infty} \frac{1}{f_nf_{n+2}}$ - diverge or converge I am currently going through exercises regarding convergence/divergence.
For my previous question I used the ratio test, and managed to get through it all okay (I think). I proved that: $$\sum_{n=1}^{\infty} \frac{n!}{n^n}$$ 
converges, and now I have to show whether or not an inverse Fibonacci sum converges/diverges and I'm not sure what method to use. What is the best way to tackle this problem?
$$\sum_{n=1}^{\infty} \frac{1}{f_nf_{n+2}}$$ 
Where $f_n$ is the Fibonacci sequence, $f_n = f_{n-1} + f_{n-2}$ with initial terms $f_1 = f_2 = 1$
I don't believe it's similar to how I completed $\sum_{n=1}^{\infty} \frac{n!}{n^n}$ but let me know if I'm wrong. Based on looking at fairly similar questions on this website I have started trying to use proof by contradiction.

Another approach. The $n$th Fibonacci number is about $\varphi^n$, where $\varphi = (1+\sqrt{5})/2$ is the golden mean. Then your sum behaves like $\Sigma (1/\varphi^{2n})$. It's easy to show that converges. Of course you don't get the value of what it converges to, as in @achillehui 's nice answer.
