$N$ 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:

• $d_7(1)=d_7(\frac{1}{2})=d_7(\frac{1}{4})=d_7(\frac{1}{5})=0$
• $d_7(\frac{1}{3})=3 \text{ since } \frac{1}{3}= 0.3333333333...$
• $d_7(\frac{1}{6})=6 \text{ since } \frac{1}{6}= 0.1666666666...$
• $d_7(\frac{1}{7})=1 \text{ since } \frac{1}{7}= 0.1428571428...$

Let $S(n)=\sum_{k=1}^nd_n\left(\cfrac{1}{k}\right)$.

You are given:

• $S(7)=0+0+3+0+0+6+1=10$
• $S(100)=418$

Find $S(10^7)$.