A harmonic series without the nines

[Thecla]

TL;DR Summary

The sum of an harmonic series without numbers containing a nine is finite

The sum of the harmonic series(1/1+1/2+1/3...) is infinite. However, if you exclude all the terms that contain the number nine, the sum is just under 23.
From 1 to 100 19% of the terms are excluded
From 1 to 1000 27.1% of the terms are excluded
Is there a formula for a N digit number what the percentage of numbers from 1 to N that contain a 9?

 

[mfb]

There are 10^N numbers with up to N digits (or exactly N digits if we use leading zeros) because every digit has 10 options. There 9^N numbers with up to N digits but no 9 for the same reason. That leads to a simple equation for the fraction of numbers with/without 9 up to N digits.