TITLE: Infinite Fibonacci sums $\sum_{n=1}^{\infty} \frac{1}{f_nf_{n+2}}$ - diverge or converge
QUESTION [9 upvotes]: 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.

REPLY [5 votes]: Since it is easy to see that $f_n\geq n$ for $n\geq5$ you can bound the (positive) summand $\frac1{f_nf_{n+2}}<\frac1{n^2}$ for $n\geq5$; therefore since $\sum_{n\geq1}\frac1{n^2}$ converges, so does $\sum_{n\geq1}\frac1{f_nf_{n+2}}$.