Kempner series

The Kempner series[1][2]: 31–33  is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum

where the prime indicates that n takes only values whose decimal expansion has no nines. The series was first studied by A. J. Kempner in 1914.[3] The series is counterintuitive[1] because, unlike the harmonic series, it converges. Kempner showed the sum of this series is less than 90. Baillie[4] showed that, rounded to 20 decimals, the actual sum is 22.92067661926415034816 (sequence A082838 in the OEIS).

Heuristically, this series converges because most large integers contain every digit. For example, a random 100-digit integer is very likely to contain at least one '9', causing it to be excluded from the above sum.

Schmelzer and Baillie[5] found an efficient algorithm for the more general problem of any omitted string of digits. For example, the sum of ⁠1/n⁠ where n has no instances of "42" is about 228.44630415923081325415. Another example: the sum of ⁠1/n⁠ where n has no occurrence of the digit string "314159" is about 2302582.33386378260789202376. (All values are rounded in the last decimal place.)

  1. ^ a b Weisstein, Eric W. "Kempner series". MathWorld.
  2. ^ Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton: Princeton University Press. ISBN 978-0-691-09983-5.
  3. ^ Kempner, A. J. (February 1914). "A Curious Convergent Series". American Mathematical Monthly. 21 (2). Washington, DC: Mathematical Association of America: 48–50. doi:10.2307/2972074. ISSN 0002-9890. JSTOR 2972074.
  4. ^ Cite error: The named reference Baillie 1979 was invoked but never defined (see the help page).
  5. ^ Schmelzer, Thomas; Baillie, Robert (June–July 2008). "Summing a Curious, Slowly Convergent Series". American Mathematical Monthly. 115 (6). Washington, DC: Mathematical Association of America: 525–540. doi:10.1080/00029890.2008.11920559. ISSN 0002-9890. JSTOR 27642532. MR 2416253. S2CID 11461182.