<s>[INST] Finding a closed form for $\sum_{n=1}^{\infty}\frac{1}{F(n)F(n+2k)}$, where $F(n)$ is the $n$-th Fibonacci number So, I want to find the closed value for the sequence
$$\sum_{n=1}^{\infty} \frac{1}{F(n)F(n+2k)}$$
Here by $F(n)$ I mean the $n$-th term of Fibonacci Sequence.
I got motivated for this by the YouTube video "Using partial fractions to evaluate two Fibonacci reciprocal sums" by Prof. Michael Penn. So in the end of video he says for finding this for the general case : k. So i tried it and here is what i did:-
Let
$$\frac{1}{F(n)F(n+2k)} = \frac{A}{F(n)} + \frac{B}{F(n+2k)}$$
so by rearranging we can say that: $AF(n+2k) + BF(n) = 1$
Now let $C_0 = F(n+2k)$
then $C_1 = F(n+2k-1) + F(n+2k-2)$
so   $C_2 = 2F(n+2k-2) + F(n+2k-3)$
then by induction we can prove that if i repeat this process j times then $C_j = F(j+1)F(n+2k-j) + F(j)F(n+2k-j-1)$
So now let $j = 2k - 1$
F
then $C_{2k-1} = F(2k)F(n+1) + F(2k-1)F(n)$
now putting this in to the Eq, of A and B then $AF(2k)F(n+1) + AF(2k-1)F(n) + BF(n) = 1$
now by putting $F(n+1) = F(n+2) - F(n)$
then I will get $AF(2k)F(n+2) + (B+AF(2K-1) - AF(2k))F(n = 1)$
there can be infinite values of A and B satisfying this relation so if I let $A = \frac{1}{F(2k)F(n+2)}$ becaues the first term in the Eq.  will become 1 then $B = \frac{F(2k-2)}{F(2k)F(n+2)}$
By putting this values in the summation I will get $\sum_{n=1}^{\infty}\frac{1}{F(2k)F(n)F(n+2)} + \frac{F(2k-2)}{F(n+2)F(n+2k)}$
now in the video It is showned that $\sum_{n=1}^{\infty}\frac{1}{F(n)F(n+2)} = 1$ so I am not going to derive it.
Now putting it we get $\frac{1 + F(2k-2)\sum_{n=1}^{\infty}\frac{1}{F(n+2)F(n+2k)}}{F(2k)}$
$\frac{1 + F(2k-2)\sum_{n=3}^{\infty}\frac{1}{F(n)F(n+2(k-1))}}{F(2k)}$ in this I have changed the limit.
Now by adding and subtracting some termes I get $Sum(k) = \frac{1 + F(2k-2)\sum_{n=1}^{\infty}\frac{1}{F(n)F(n+2(k-1))} - \frac{F(2k-2)}{F(2k-1)} - \frac{F(2k-2)}{F(2k)}}{F(2k)}$
Now finnaly I ended up with this $Sum(k) = \frac{1 + F(2k-2)Sum(k-1) - \frac{F(2k-2)}{F(2k-1)} - \frac{F(2k-2)}{F(2k)}}{F(2k)}$
But this is not a direct formulae. Can someone help?
 [/INST] After revising your work, you should find
$$\sum_{n=1}^{\infty} \frac{1}{F_nF_{n+2k}}=\frac{1}{F_{2k}}\sum_{j=1}^k\frac{1}{F_{2j-1}F_{2j}}.$$
The first terms for $k\geq 1$ are $1$, $\frac{7}{18}$, $\frac{143}{960}$, $\frac{4351}{76440}$, $\frac{814001}{37437400}$ (see also OEIS sequence A333088).
Proof. By induction with respect to $k\geq 1$, we verify that
$$\frac{F_{2k}}{F_nF_{n+2k}}=\sum_{j=1}^k\frac{1}{F_{n+2(j-1)}F_{n+2j}}.$$
Indeed, it holds for $k=1$ and, for the inductive step, we have
$$\sum_{j=1}^{k+1}\frac{1}{F_{n+2(j-1)}F_{n+2j}}
=\frac{F_{2k}}{F_nF_{n+2k}}+\frac{1}{F_{n+2k}F_{n+2k+2}}=\frac{F_{2k+2}}{F_nF_{n+2k+2}}.$$
where at the last step we applied Vajda's identity: $$F_{2k+2}F_{n+2k}-F_{2k}F_{n+2k+2}=(-1)^{2k}F_2F_n=F_n.$$
Hence, since $\sum_{n=1}^{\infty} \frac{1}{F_nF_{n+2}}=1$ (watch the video or see Sum $\frac{1}{1\times2}+\frac{1}{1\times3}+\frac{1}{2\times5}+\frac{1}{3\times8}+\cdots$), it follows that
\begin{align}
\sum_{n=1}^{\infty} \frac{1}{F_nF_{n+2k}}&=
\frac{1}{F_{2k}}\sum_{n=1}^{\infty}\sum_{j=1}^k\frac{1}{F_{n+2(j-1)}F_{n+2j}}\\
&=\frac{1}{F_{2k}}\sum_{j=1}^k\sum_{n=1}^{\infty}\frac{1}{F_{n+2(j-1)}F_{n+2j}}=\frac{1}{F_{2k}}\sum_{j=1}^k\left(1-\sum_{n=1}^{2(j-1)}\frac{1}{F_{n}F_{n+2}}\right)\\
&=\frac{1}{F_{2k}}\sum_{j=1}^k\left(1-\sum_{n=1}^{2(j-1)}\left(\frac{1}{F_{n}F_{n+1}}-\frac{1}{F_{n+1}F_{n+2}}\right)\right)\\
&=\frac{1}{F_{2k}}\sum_{j=1}^k\left(1-1+\frac{1}{F_{2j-1}F_{2j}}\right)
=\frac{1}{F_{2k}}\sum_{j=1}^k\frac{1}{F_{2j-1}F_{2j}}.
\end{align}
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