Partial sums of the Kempner series The Kempner series is a series that sums the inverse of all positive integers that don't contain a &quot;9&quot; in their base-10 representations (i.e., \$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + .. + \frac{1}{8} + \frac{1}{10} + ...\$).
It can be shown that, unlike the Harmonic series, the Kempner series converges (to a value of about 22.92067661926415034816).
Your task is to find the partial sums of the Kempner series. These are the ways you can do it:

Take a number \$n\$, and return the sum of the inverse of the first \$n\$ numbers that don't have a &quot;9&quot; in them, which is the \$n\$th partial sum of the series.
Take a number \$n\$, and return the first \$n\$ partial sums of the series.
Don't take any input and output the partial sums infinitely.

You can choose if your input is 0-indexed or 1-indexed.
Your algorithm result's distance from the correct value may not be over \$10^{-4}\$, for all possible values of \$n\$. While your algorithm should work theoretically for all values for N, you may ignore inaccuracies coming from floating-point errors.
Test cases, in case of returning the \$n\$th partial sum, 0-indexed:
0 -&gt; 1.0
1 -&gt; 1.5
9 -&gt; 2.908766...
999 -&gt; 6.8253...

Standard loopholes are disallowed.
This is code-golf, so the shortest answer in bytes wins.
Jelly, 7 6 bytes
Rb9ḌİS

Try it online!
-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