Helo,

Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving

$$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\right\} =1-\log\left(\frac{1+\sqrt{5}}{2}\right)$$

A lecturer wrote this formula on the board without reference and without proof! Thank you for any hint.

Helo,

Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving

$$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\right\} =1-\log\left(\frac{1+\sqrt{5}}{2}\right)$$

A lecturer wrote this formula on the board without reference and without proof! Thank you for any hint.

ps: the proposed limit is wrong. See the answer below which proves that the limit does indeed exist but has value $3\log(2)/4$.