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 $\frac{1}{3}=$ 0.3333333333...
• $d_7\left(\frac{1}{6}\right)=6$ since $\frac{1}{6}=$ 0.1666666666...
• $d_7\left(\frac{1}{7}\right)=1$ since $\frac{1}{7}=$ 0.1428571428...

Let ${\displaystyle S(n)=\sum _{k=1}^n{d}_n\left(\frac{1}{k}\right)}$.

You are given:

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

Find $S({10}^7)$.