Problem 970

Kangaroo Hopping over Sixes

Starting at zero, a kangaroo hops along the real number line in the positive direction. Each successive hop takes the kangaroo forward a uniformly random distance between $0$ and $1$. Let $H(n)$ be the expected number of hops needed for the kangaroo to pass $n$ on the real line.

For example, $H(2) \approx 4.67077427047$. The first eight digits after the decimal point that are different from six are $70774270$.

Similarly, $H(3) \approx 6.6665656395558899$. Here the first eight digits after the decimal point that are different from six are $55395558$.

Find $H(10^6)$ and give as your answer the first eight digits after the decimal point that are different from six.

袋鼠跳过六

一只袋鼠从原点出发，沿实数轴正方向跳跃，每次跳跃的距离是一个在$0$到$1$之间均匀分布的随机数。记$H(n)$为袋鼠在实数轴上跳跃的总距离超过整数$n$所需的期望跳跃次数。

例如，$H(2) \approx 4.67077427047$，小数点后前八个不是六的数字为$70774270$。

类似地，$H(3) \approx 6.6665656395558899$，小数点后前八个不是六的数字为$55395558$。

求$H(10^6)$，并给出小数点后前八个不是六的数字作为你的答案。