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Let $\mu_m=\sum_{k=0}^{\left \lceil{m/2}\right \rceil }k\frac{{m-k+1}\choose k}{f_{m+2}}$ where $f_{m+2}$ is the $m+2$th Fibonacci number with $f_1=f_2=1$.

Then $\lim_{m\to\infty}\frac{\mu_m}{m}=(5-\sqrt5)/10$.

So my textbook solutions said to use Mathematica to find this limit. I'm guessing that this uses some pretty advanced mathematics. I would still like to at least have a rough idea of the way to get this limit. Thanks

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    $\begingroup$ Did you mean $f_{m+2}$ or $f_{k+2}$? Since you are summing over $k$. $\endgroup$
    – Math1000
    Jan 17, 2020 at 1:09
  • $\begingroup$ @Math1000 My book says $f_{m+2}$. Is it supposed to be $f_{k+2}$? $\endgroup$
    – johnson
    Jan 17, 2020 at 12:32
  • $\begingroup$ It just seems weird to put $f_{m+2}$ within the sum when $f_{m+2}$ is not a function of the summation index. $\endgroup$
    – Math1000
    Jan 17, 2020 at 12:50

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