Sum of Reciprocals in Base 10 with Zeroes Removed

Theorem
The infinite series


 * $\displaystyle \sum_{P \left({n}\right)} \dfrac 1 n$

where $P \left({n}\right)$ is the propositional function:
 * $\forall n \in \Z_{>0}: P \left({n}\right) \iff$ the decimal representation of $n$ contains no instances of the digit $0$

converges to the approximate limit $23 \cdotp 10345 \ldots$