Possible Comment on A007095

[David Wilson]

Possible Comment on A007095The straightforward proof that SUM_{n has no 9} 1/n < 80 is of course to show that for d >= 1, there are 8*9^(d-1) d-digit numbers with no 9. The smallest d-digit number is 1/10^(d-1), so the sum of 9-less d-digit numbers is bounded above by 8*(9/10)^(d-1). Summing this for d >= 1 gives 80. The argument would have been exactly the same for any positive digit (though possibly slightly different for 0, since 0 cannot be a leading digit, which lowers the count of 0-less d-digit numbers). Hence 80 is an upper bound for the sum of the reciprocals of d-less numbers for any 1 <= d <= 9.

----- Original Message ----- 
  From: Rob Pratt 
  To: Jeremy Gardiner ; sequence fans 
  Sent: Monday, May 21, 2007 9:12 PM
  Subject: RE: Possible Comment on A007095


  The latest issue (May 2007) of The College Mathematics Journal has a "Classroom Capsule" that generalizes this result.

  "On the Convergence of Some Modified p-Series" by Dongling Deng



------------------------------------------------------------------------------
  From: Jeremy Gardiner [mailto:jeremy.gardiner at btinternet.com] 
  Sent: Monday, May 21, 2007 2:49 AM
  To: sequence fans
  Subject: Possible Comment on A007095


  Dear seqfans,

  I saw the following at Jeremy Rouse's maths problems web page:

  http://www.math.wisc.edu/~rouse/problems.html 

  Problem 10.  Let S be the set of positive integers that, when written in base 10, does not contain the digit 9. Show that the sum of 1/n over all n ' S converges and is less than 80. (Problem 157, USSR Olympiad Problem Book). 

  This could make an interesting comment on A007095 (Numbers in base 9, also numbers without 9 as a digit), but can anyone confirm the reference?

  My guess from a google search is this may be from "The USSR Olympiad Problem Book : Selected Problems and Theorems of Elementary Mathematics" by D. O. Shklarsky, N. N. Chentzov, and I. M. Yaglom (Paperback - Sep 28 1993), however I do not possess a copy.

  Jeremy Gardiner



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