Problem 820

$N$^th digit of Reciprocals

Let $d_n(x)$ be the $n$^th decimal digit of the fractional part of $x$, or $0$ if the fractional part has fewer than $n$ digits.

For example:

$d_7 \left( 1 \right) = d_7 \left( \frac 1 2 \right) = d_7 \left( \frac 1 4 \right) = d_7 \left( \frac 1 5 \right) = 0$
$d_7 \left( \frac 1 3 \right) = 3$ since $\frac 1 3 = 0.333333{\color{red}3}333\ldots$
$d_7 \left( \frac 1 6 \right) = 6$ since $\frac 1 6 = 0.166666{\color{red}6}666\ldots$
$d_7 \left( \frac 1 7 \right) = 1$ since $\frac 1 7 = 0.142857{\color{red}1}428\ldots$

Let $\displaystyle S(n) = \sum_{k=1}^n d_n \left( \frac 1 k \right)$.

You are given:

Find $S(10^7)$.

记$d_n(x)$为$x$的小数点后第$n$位数字；若其小数部分不满$n$位则记为$0$。

例如：

$d_7 \left( 1 \right) = d_7 \left( \frac 1 2 \right) = d_7 \left( \frac 1 4 \right) = d_7 \left( \frac 1 5 \right) = 0$
$d_7 \left( \frac 1 3 \right) = 3$，因为$\frac 1 3 = 0.333333{\color{red}3}333\ldots$
$d_7 \left( \frac 1 6 \right) = 6$，因为$\frac 1 6 = 0.166666{\color{red}6}666\ldots$
$d_7 \left( \frac 1 7 \right) = 1$，因为$\frac 1 7 = 0.142857{\color{red}1}428\ldots$

记$\displaystyle S(n) = \sum_{k=1}^n d_n \left( \frac 1 k \right)$。

已知：

求$S(10^7)$。