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.