An Interesting Series Sequel

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.

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