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Problem 965

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Expected Minimal Fractional Value

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.

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最小分数期望值

记$\{x\}$为实数$x$的小数部分。

定义$f_N(x)$为，对于所有满足$0 < n \le N$的整数$n$，$\{nx\}$的最小值。

进一步定义$F(N)$为，当$x$在$[0, 1]$上均匀随机选择时，$f_N(x)$的期望值。

已知$F(1) = \frac{1}{2}$，$F(4) = \frac{1}{4}$，$F(10) \approx 0.1319444444444$。

求$F(10^4)$，并四舍五入到小数点后$13$位作为你的答案。

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