Eccoci.

Let $\{x\}$ denote the fractional part of a real number $x$.

Define $f_N(x)$ to be the minimal value of $\{nx\}$ for integer $n$ satisfying $0 < n \le N$.
Further define $F(N)$ to be the expected value of $f_N(x)$ when $x$ is sampled uniformly in $[0, 1]$.

You are given $F(1) = \frac{1}{2}$, $F(4) = \frac{1}{4}$ and $F(10) \approx 0.1319444444444$.

Find $F(10^4)$ and give your answer rounded to 13 digits after the decimal point.