Kempner series

In mathematics , the ten Kempner series , named after Aubrey J. Kempner , denote those series that are created by removing all summands from the harmonic series that contain a certain decimal number in their denominator. The Kempner series therefore belong to the subharmonic series . ${\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}}$

If one omits all summands whose denominator contains the digit in its decimal notation, the Kempner series results as ${\displaystyle 0}$ ${\displaystyle K_{0}}$

${\displaystyle K_{0}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{9}}+{\frac {1}{11}}+\cdots +{\frac {1}{19}}+{\frac {1}{21}}+\cdots +{\text{ usw. }}+\cdots +{\frac {1}{99}}+{\frac {1}{111}}+\ldots }$

Or by leaving out the summands with one in the denominator: ${\displaystyle 1}$

${\displaystyle K_{1}={\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{9}}+{\frac {1}{20}}+{\frac {1}{22}}+\cdots +{\frac {1}{30}}+{\frac {1}{32}}+\cdots +{\text{ usw. }}+\cdots +{\frac {1}{99}}+{\frac {1}{200}}+{\frac {1}{202}}+\cdots }$

They were first described by Aubrey J. Kempner in 1914.

The interesting thing about these ten series is that they all converge , although the harmonic series itself does not converge. This was proven by Kempner; therefore the series are often called the Kempner series. The convergence property is also clear from the fact that already from 7-digit numbers these are mostly omitted and with large numbers there are only a few who do not contain a certain number and can thus make an addition contribution.

Proof of Convergence

For the Kempner series are ${\displaystyle K_{0}}$

• in the single-digit denominator range 1 to 9 exactly denominators (all) permitted; ${\displaystyle \!\ 9}$
• in the two-digit denominator range 10 to 99 exact denominators (nine digits in the first position times nine digits in the second position possible); ${\displaystyle 9\cdot 9=9^{2}}$
• in the three-digit denominator range 100 to 999 exactly denominators permitted; etc., ${\displaystyle 9\cdot 9\cdot 9=9^{3}}$

are general

• in the -digit denominator range up to the exact denominator. ${\displaystyle n}$ ${\displaystyle 10^{n-1}}$ ${\displaystyle 10^{n}-1}$ ${\displaystyle \!\ 9^{n}}$

The permissible single-digit denominator values ​​are all greater than or equal to 1, so the fractions in the series are each less than or equal to 1; the permissible two-digit denominators are all greater than or equal to 10, so the corresponding fractions are all less than or equal to ; the three-digit permissible denominators are each greater than or equal to 100, so the corresponding fractions are all less than or equal to ; etc. ${\displaystyle 9}$ ${\displaystyle 9^{2}}$ ${\displaystyle {\tfrac {1}{10}}}$ ${\displaystyle 9^{3}}$ ${\displaystyle {\tfrac {1}{100}}}$

That gives the upper bound

${\displaystyle {\begin{matrix}K_{0}&=&({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{9}})&+&({\frac {1}{11}}+\cdots +{\frac {1}{99}})&+&({\frac {1}{111}}+\cdots +{\frac {1}{999}})&+&\cdots \\&<&({\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1}}+\cdots +{\frac {1}{1}})&+&({\frac {1}{10}}+\cdots +{\frac {1}{10}})&+&({\frac {1}{100}}+\cdots +{\frac {1}{100}})&+&\cdots \\&=&9\cdot {\frac {1}{1}}&+&9^{2}\cdot {\frac {1}{10}}&+&9^{3}\cdot {\frac {1}{100}}&+&\cdots \end{matrix}}}$

${\displaystyle =9\cdot \left(1+\left({\frac {9}{10}}\right)+\left({\frac {9}{10}}\right)^{2}+\left({\frac {9}{10}}\right)^{3}+\cdots \right)}$

${\displaystyle ={\frac {9}{1-{\frac {9}{10}}}}=90.}$

(The row in the penultimate row is a convergent geometric row )

This converges and the (rather generous) bound applies ${\displaystyle K_{0}}$

${\displaystyle K_{0}<90.}$

The proof of the convergence of the other series is analogous, but it should be noted that only 8 values are permitted in the one-digit denominator range, but denominator values ​​in the two-digit denominator range, since both the zero and the corresponding digit are in the first position and in the second but only the corresponding number "forbidden" are etc .; overall this results in the limit . ${\displaystyle 8\cdot 9}$ ${\displaystyle 80}$

values

The rows converge extremely slowly.

Approximate values

Omitted digit	Approximate value
0	23.10344
1	16.17696
2	19.25735
3	20.56987
4th	21.32746
5	21.83460
6th	22.20559
7th	22.49347
8th	22.72636
9	22.92067

Efficient calculation options

Because of the rather slow convergence, fast and efficient calculation algorithms are required, cf.

Extensions

n times

F. Irwin generalized the result of the convergence of the ten Kempner series by proving that all series, which over the reciprocal values ​​of all natural numbers, in which the digit occurs exactly , the digit exactly , etc., also converge. ${\displaystyle x_{0}}$ ${\displaystyle n_{0}}$ ${\displaystyle x_{1}}$ ${\displaystyle n_{1}}$

The sum of the reciprocal values ​​of the natural numbers that contain exactly a 9 is about 23.044287080747848319. This value is larger than Kempner's , although it begins with larger summands. A more extreme example of this is the sum of the reciprocal values ​​of the natural numbers, in which one hundred zeros occur, it starts with the summand and is still greater than approximately . ${\displaystyle K_{9}}$ ${\displaystyle 1/10^{100}}$ ${\displaystyle K_{9}}$

Related sequences of digits

One possibility to thin out the harmonic series far less is to only take out all summands whose denominator contains a certain connected sequence of digits somewhere - about 314 (the first three digits of the circle number ${\displaystyle \pi }$ ). Such series also converge; in the example mentioned, the limit value is about 2299.829782. Removing the first six digits 314159 results in a limit value of around 2302582.333863782607892. In general, the following applies: If all summands with a coherent sequence of digits of length are removed, the series converges with a limit value in the order of magnitude of approximately . ${\displaystyle n}$ ${\displaystyle 10^{n}\cdot \ln 10}$

In other place value systems

There are of course also analog series in other place value systems . The dual Kempner series, for example, is created by deleting all summands that contain one in their dual representation. It is not possible to delete all binary numbers in which one occurs. So the only dual Kempner series is ${\displaystyle {\rm {O}}}$ ${\displaystyle {\rm {I}}}$

${\displaystyle {\begin{alignedat}{2}K_{\text{dual}}&={\frac {\rm {I}}{\rm {I}}}+{\frac {\rm {I}}{\rm {II}}}+{\frac {\rm {I}}{\rm {III}}}+{\frac {\rm {I}}{\rm {IIII}}}+\cdots &{\text{(dual)}}\\&=1+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{15}}+{\frac {1}{31}}+\cdots &{\text{ (dezimal)}}\\&=\sum _{k=1}^{\infty }{\frac {1}{2^{k}-1}}\\&=1{,}60669515241529\ldots ,\end{alignedat}}}$

which converges to the Erdős-Borwein constant . To prove the convergence, consider the infinite convergent geometric series as the upper bound. ${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{2^{k}}}=\sum _{k=0}^{\infty }\left({\frac {1}{2}}\right)^{k}={\frac {1}{1-{\frac {1}{2}}}}=2}$

literature

• Julian Havil : Gamma: Euler's constant, prime number beaches and the Riemann hypothesis . Springer, Berlin 2007, p. 42ff. ISBN 978-3-540-48495-0
• Sequence A082839 in OEIS and sequence A082830 in OEIS

Individual evidence

1. ^ Aubrey J. Kempner: A Curious Convergent Series . In: Amer. Math. Monthly , Vol. 21 No. 2, Mathematical Association of America, Washington 1914, pp. 48-50, ISSN  0002-9890 .
2. ↑ Note: Probability of occurrence of a digit in an n-digit decimal group of digits: P (n) = 1 - (9/10) ^ n. For n = 7: P> 50%.
3. ↑ Eric W. Weisstein : Kempner Series . In: MathWorld (English).
4. ^ ^{A b} Robert Baillie: Summing the Curious Series of Kempner and Irwin , June 27, 2008, arxiv
5. ^ F. Irwin: A Curious Convergent Series . In: Amer. Math. Monthly. Volume 23, 1916, pages 149-152.
6. ^ R. Baillie, T. Schmelzer: Summing Kempner's Curious (Slowly-Convergent) Series. May 20, 2008; see. in Wolfram Library Archive
7. ^ R. Baillie, T. Schmelzer: Summing Kempner's Curious (Slowly-Convergent) Series. May 20, 2008; see. in Wolfram Library Archive
8. ↑ Eric W. Weisstein : Kempner Series . In: MathWorld (English).