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:

$d_{7} (1) = d_{7} (\frac{1}{2}) = d_{7} (\frac{1}{4}) = d_{7} (\frac{1}{5}) = 0$
$d_{7} (\frac{1}{3}) = 3$ since $\frac{1}{3} = 0.333333 3333 \dots$
$d_{7} (\frac{1}{6}) = 6$ since $\frac{1}{6} = 0.166666 6666 \dots$
$d_{7} (\frac{1}{7}) = 1$ since $\frac{1}{7} = 0.142857 1428 \dots$

Let $S (n) = \sum_{k = 1}^{n} d_{n} (\frac{1}{k})$ .

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：

$d_{7} (1) = d_{7} (\frac{1}{2}) = d_{7} (\frac{1}{4}) = d_{7} (\frac{1}{5}) = 0$
$d_{7} (\frac{1}{3}) = 3$ ，因为 $\frac{1}{3} = 0.333333 3333 \dots$
$d_{7} (\frac{1}{6}) = 6$ ，因为 $\frac{1}{6} = 0.166666 6666 \dots$
$d_{7} (\frac{1}{7}) = 1$ ，因为 $\frac{1}{7} = 0.142857 1428 \dots$

记 $S (n) = \sum_{k = 1}^{n} d_{n} (\frac{1}{k})$ 。

已知：

求 $S (10^{7})$ 。