An Interesting Series Sequel

Number Theory Level 4

In the infinite series $\dfrac{1}{9}+\dfrac{1}{99}+\cdots + \dfrac{1}{10^n-1}+\cdots$ at what place after the decimal point does the second $1$ occur?

$\text{Details and Assumptions}$

The first $1$ occurs at the first place after the decimal point.