 | 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 whose positive integer
denominators contain the decimal digit 
, where 
, 1, ..., 9, 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 decimal
digits of 
contain no string 314. This sum has approximate value 2299.829782.... The approximation
for an excluded string of length
depends on the pattern of the string.
For example, if the excluded string consists of 
repetitions of one decimal digit,
the sum divided by 
approaches 
as 
,
rather than 
(Schmelzer and Baillie 2008).
