On 3/29/11 10:25 PM, James Van Buskirk wrote: > "Dan Hoey"<email@example.com> wrote in message > news:firstname.lastname@example.org... > >> On 3/25/2011 12:25 AM, James Van Buskirk wrote: >>> Given the O.P. I was hoping for something like the sum of the >>> reciprocals of all the positive integers that don't have the digit >>> '8'. > >> I like this more than Leroy Quet's puzzle. Do you have any ideas >> about how to calculate this number? > > I calculated up to double or quadruple precision once. The version > with 9 as the missing digit was from the Olympiad problem book and > you can start with the proof that the series converges and apply > a little more effort and get to something you can evaluate. No one > has ever given an independent calculation that I can check against, > however.
According to http://oeis.org/A082837, the sum of reciprocals of positive integers that don't have the digit 8 is 22.726365402679370602833644156742557889210702616360219843536376162. For those without the digit 9, http://oeis.org/A082838 has 22.920676619264150348163657094375931914944762436998481568541998356. Robert Baillie has a paper at http://arxiv.org/abs/0806.4410v2 that covers these and related topics and includes Mathematica code.