 | sum | OEIS |
| 0 | 23.10344 | A082839 |
| 1 | 16.17696 | A082830 |
| 2 | 19.25735 | A082831 |
| 3 | 20.56987 | A082832 |
| 4 | 21.32746 | A082833 |
| 5 | 21.83460 | A082834 |
| 6 | 22.20559 | A082835 |
| 7 | 22.49347 | A082836 |
| 8 | 22.72636 | A082837 |
| 9 | 22.92067 | A082838 |
A Kempner series 
is a series obtained by removing all terms containing a single digit 
from the harmonic series.
Surprisingly, while the harmonic series diverges,
all 10 Kempner series converge. For example,
 |
While they are difficult to calculate, the above table summarizes their approximate values as computed by Baillie (1979; Havil 2003, pp. 33-34).
Schmelzer and Baillie (2008) have devised an improved algorithm for summing more general Kempner series, such as the sum of 
where the digits of 
contain no string 314. This sum has approximate value 
. In general, the 
when a particular
string of length 
is excluded from the 
's
summed over is approximately given by 
(Baillie and Schmelzer 2008).
