Partial sums of the Kempner series
Jelly, 7 6 bytes
Rb9ḌİS

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-1 byte thanks to @Razetime
Takes input 1-indexed as an argument (footer on TIO converts to 0-indexed like in test cases)
Returns the nth partial sum
How it Works
Let's look at the denominators in base 10:
\$[1, 2, 3, 4, 5, 6, 7, 8, 10,11,12,13,...]\$ (base 10)
Since these consist of the nine base-9 digits, we can interpret them as  base 9 strings:
\$[1_9, 2_9, 3_9, 4_9, 5_9, 6_9, 7_9, 8_9, 10_9, 11_9, 12_9, 13_9, ...]\$
\$=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ]\$ (base 10)
So all of the denominators can be generated as natural numbers, converted to base 9, then interpreted as base 10.
Rb9ḌİS - main link, taking n (1-indexed)
R       - convert to list [1..n]
 b9     - convert each natural number to base 9
   Ḍ   - interpret each base-9 string as base 10 (decimal)
    İ  - reciprocal
     S - sum