Kempner series

In mathematics denote the ten Kempner series, named after Aubrey J. Kempner, those rows that arise that you removed from the harmonic series all summands that contain a specific decimal digit in its denominator. Therefore, the Kempner series belong to the sub-harmonic series.

Leaving about all summands away, the denominator contains the digit in its decimal form, the Kempner - series results as

Or by omitting the summands with a in the denominator:

They were first described by Aubrey J. Kempner 1914.

The interesting thing about these ten rows is that they all converge, although the harmonic series does not converge even. This was proved by Kempner; hence the series are often called Kempner series.

  • 3.1 n-fold occurrence
  • 3.2 Coherent sequence of digits
  • 3.3 In other value systems

Proof of convergence

For the Kempner series

  • Digit denominator range 1 to 9 exactly denominator (all) allowed;
  • Double-digit denominator range 10 to 99 exactly denominator ( nine digits in the first place by nine digits in the second position possible) allowed;
  • Triple-digit denominator range 100-999 exactly denominator permitted; etc.,

Generally are

  • In - digit denominator range up exactly denominator allowed.

The allowable digit denominator values ​​are all greater than or equal to 1, so the breaks in the series are each less than 1; the allowed double-digit denominators are all greater than or equal 10, therefore, the corresponding fractions are all less than or equal; the three-digit allowable denominator are each greater or equal to 100, therefore, the corresponding fractions are all less than or equal; etc.

( In the series in the penultimate line is a convergent geometric series )

This converges and it is the ( rather generous ) barrier

The proof of the convergence of the other rows is analogous, but it is to note that in the single-digit denominator area only 8 values ​​in the two -digit denominator area but denominator values ​​are allowed, as in the first position, both the zero and the corresponding digit in the second position but "forbidden" only the corresponding number are, etc.; total this results in the barrier.


The series converge extremely slowly.


Efficient calculation options

Due to the rather slow convergence one needs fast and efficient calculation algorithms, cf.


N-fold occurrence

F. Irwin generalized the result of the convergence of the ten Kempner - series by proving that all rows that also converge on the reciprocals of all natural numbers in which the digit exactly once, the number exactly, etc. occur.

The sum of the reciprocals of the natural numbers in which occurs exactly a 9, is about 23.044287080747848319. This value is greater than Kempner, although it starts with larger addends. A more extreme example is the sum of the reciprocals of natural numbers, in which there are one hundred and zeros, starting with the summands and is still greater than about.

Related digits

One way to thin out the harmonic series is far less, take out only every summand whose denominator somewhere a certain contiguous sequence of numbers - about 314 (the first three digits of the mathematical constant ) - contains. Also, such series converge; in the example, there is a limit of about 2299.829782. When taking out the first six digits of 314159 there is a limit of about 2,302,582.333863782607892. In general, if all the summands are removed with a continuous number sequence length, the series converges to a limit value in the order of about.

In other value systems

There are of course analogous series in different value systems. The dual Kempner series as produced by brushing all summands which contain a in their dual representation. To delete all binary numbers in which a occurs, do not go. The only dual Kempner series is therefore

Which converges to the Erdős - Borwein constant. To prove the convergence, consider the infinite convergent geometric series as an upper bound.


  • Julian Havilland: Gamma: Euler's constant, prime beaches and the Riemann Hypothesis. Springer, Berlin 2007, pp. 42ff. ISBN 978-3-540-48495-0
  • Sequence A082839 in OEIS A082830 in OEIS sequence and
  • Sequences and series