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.
Best Answer
Notice $$\frac{1}{f_n f_{n+2}} = \frac{f_{n+1}}{f_nf_{n+1}f_{n+2}} =\frac{f_{n+2}-f_{n}}{f_n f_{n+1} f_{n+2}} = \frac{1}{f_nf_{n+1}}-\frac{1}{f_{n+1}f_{n+2}}$$ We are dealing with a telescoping sum and
$$\sum_{n=1}^\infty \frac{1}{f_nf_{n+2}} = \lim_{p\to\infty} \sum_{n=1}^p \frac{1}{f_nf_{n+2}} = \lim_{p\to\infty}\left(\frac{1}{f_1f_2} - \frac{1}{f_{p+1}f_{p+2}}\right) = \frac{1}{f_1 f_2} = 1$$