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I found this problem from my old note, with the memo 'The answer is very interesting."
But I forgot how to solve it!
I rummaged through all my bookshelf, but I couldn't find any clue.
The only thing I remember is

  1. Partial fraction decomposition is used.
  2. It would be easier to solve by using $\sum_{n=1}^\infty \frac{1}{F_n F_{n+2}}=\sum_{n=1}^\infty \left( \frac{1}{F_n F_{n+1}} - \frac{1}{F_{n+1} F_{n+2}} \right) = 1$.
  3. The solution is not that complicated.

Would you help me?

okw1124
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2 Answers2

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It can be reduced to $\sum \frac1{F_n F_{n+2}}$ as follows.

Multiply the numerator and denominator by $F_{n+2}$, and write this in terms of $F_n$ and $F_{n+4}$ by using that the sequences $(F_{2n})$ and $(F_{2n+1})$ are linear recurrences.

You will obtain something like

$$\frac1{F_{n}F_{n+4}} = \frac{F_{n+2}}{F_{n}F_{n+2}F_{n+4}} = \frac{a}{F_nF_{n+2}} + \frac b{F_{n+2} F_{n+4}}$$ so that we obtain roughly $(a+b) \sum \frac1{F_n F_{n+2}}$.

Details:

In fact, $F_{n+4} = c_1 F_{n+2} + c_2 F_n$ with $c_1$ and $c_2$ such that the roots of the polynomial $x^2 - c_1 x - c_2$ are $\varphi^2$ and $\overline{\varphi}^2 = \varphi^{-2}$. In particular:

  • $c_1 = \varphi^2 + \varphi^{-2} = (\varphi - \varphi^{-1})^2 - 2 = (\sqrt 5 F_1)^2 - 2 = 3$;
  • $c_2 = -1$.

We obtain $$\begin{align*} \sum_{n=1}^\infty \frac1{F_n F_{n+4}} &= \sum_{n=1}^\infty \frac{F_{n+2}}{F_n F_{n+2} F_{n+4}} \\ &= \sum_{n=1}^\infty \frac{\frac1{c_1} F_{n+4} - \frac{c_2}{c_1} F_{n+1}}{F_n F_{n+2} F_{n+4}} \\ &= \frac13 \sum_{n=1}^\infty \left(\frac{1}{F_n F_{n+2}} + \frac{1}{F_{n+2} F_{n+4}} \right) \\ &= \frac23 \sum_{n=1}^\infty \frac{1}{F_n F_{n+2}} - \frac13 \frac1{F_1 F_3} - \frac13 \frac1{F_2 F_4} \\ &= \frac23 - \frac16 - \frac19 \\ &= \frac7{18} \,. \end{align*}$$

Bart Michels
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These are only hints:

  • $F_{n+4}=2F_{n}+3F_{n+1}$
  • $\frac{1}{a\cdot b}=\frac{1}{a+b}\cdot\left(\frac{1}{a}+\frac{1}{b}\right)$
  • $F_{n}+F_{n+4}=3\left(F_{n}+F_{n+1}\right)=3F_{n+2}$
  • $\frac{1}{a\cdot b}=\frac{1}{b-a}\cdot\left(\frac{1}{a}-\frac{1}{b}\right)$
  • $F_{n+2}-F_{n}=F_{n+1}$
  • Telescoping
acat3
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