I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right) ^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3)(*)$$  where $\lfloor x \rfloor$ is th floor function.

The Fibonacci sequence is defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}~(n\ge 2)$. Then we can get $$F_n=\dfrac{\alpha ^n-\beta ^n}{\sqrt{5}}$$ 
where $\alpha=\dfrac{1+\sqrt{5}}{2}$ and $\beta=\dfrac{1-\sqrt{5}}{2}.$
 
The following are some of my attempts:

> For  some example:
> 
> $n=3$, the left hand is $5$, the right hand is $5.$ 
> 
> $n=4$, the left hand is $13$, the right hand is $13.$ 
> 
> $$\vdots$$
> 
> $n=15$, the left hand is $514229$, the right hand is $514229.$ 
> 
> It is all true. But as $n$ increases, the order of magnitude grows
> very rapidly.
> 
> I ask one of my good friends to use a Python program to check $(*).$ 
> He says it is true for $n\le 35$. When $n=36$, the Python says it is not true, But when $n= 37$, it is true again.
> 
> Thus I change one way and  I ask my fiend to use a Python program to
> check $$\left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right) ^{-1}
> =F_{2n-1}~~(n\ge 3)(**).$$
> 
> Then the program shows it is true at least for $31\le n\le 51.$
>
>But as you see, the left hand  of $(**)$ is a decimal and the right hand  of $(**)$ is an integer.

So I do not know if it is because the order of magnitude on the left hand of $(*)$ is growing very fast, $(*)$ becomes not true due to some computer shortcomings.

Finally I wonder if $(*)$ is true or false? Any help and references are greatly appreciated.

Thanks!

I have also posted it on MSE as [On the finite sum of reciprocal Fibonacci sequences][1]


  [1]: https://math.stackexchange.com/questions/4747937/on-the-finite-sum-of-reciprocal-fibonacci-sequences