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I am looking for the sum of the series:

$$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$$

where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the case of $\dfrac{1}{n(n+1)}$, but do not see how it should work in this particular case. Thank you.

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\begin{align}\sum_{n=1}^{N}\frac{1}{f(n)f(n+2)} &=\sum_{n=1}^{N}\frac{\color{red} {f(n+1)}}{f(n)\color{red} {f(n+1)}f(n+2)}\\ &=\sum_{n=1}^{N}\frac{f(n+2)-f(n)}{f(n)f(n+1)f(n+2)}\\ &=\sum_{n=1}^{N}\frac{1}{f(n)f(n+1)}-\frac{1}{f(n+1)f(n+2)}\\ &=\frac{1}{f(1)f(2)}-\frac{1}{f(N+1)f(N+2)}\end{align}.

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  • $\begingroup$ This looks copied and pasted from somewhere. You should cite your sources. $\endgroup$ – David H Jun 5 '14 at 12:19
  • $\begingroup$ no I did it by my self . but my latex don't work here . it was an easy problem to me . $\endgroup$ – Khosrotash Jun 5 '14 at 12:21
  • $\begingroup$ Thank you "david H' for editing may latex codes. $\endgroup$ – Khosrotash Jun 5 '14 at 12:33
  • $\begingroup$ Seems like the main reason your latex wasn't working was that you forgot the dollar signs. I edited your response to make your latex code compatible with our site. $\endgroup$ – David H Jun 5 '14 at 12:34

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