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How to calculate this sum?

  1. Mar 5, 2008 #1
    [tex]\sum_{n=1}^{\infty} \frac{o(2^n)}{2^n} = \frac{1}{9} [/tex]
    where [tex]o(2^n)[/tex] is the number of odd digits of [tex]2^n[/tex].

    Found it in
    http://mathworld.wolfram.com/DigitCount.html
    equation (9)
     
  2. jcsd
  3. Mar 10, 2008 #2

    uart

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    Science Advisor

  4. Mar 10, 2008 #3
    Curses, JSTOR!
     
  5. Mar 14, 2008 #4
    Fantastic! Thanks a lot!
    Also found it in
    Experimentation in Mathematics: Computational Paths to Discovery
    By Jonathan M. Borwein, David H. Bailey, Roland
    pag 14-15

    here's a google link:
    http://books.google.com/books?id=cs...over&sig=yE9mO3b-YA9lLjAq6Nt4ED4bn1g#PPA15,M1

    I'm no mathematician, but found really interesting some of these strange number properties.
     
  6. Mar 15, 2008 #5

    uart

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    Thanks for the link Zhivago. Yes that book provides a nice accessible proof of that summation. In the link I posted they only really hinted at how that series was handled but in your link they nail it (only really needing knowledge of geomeric series and modolu athrithmetic to follow it). Good stuff!
     
    Last edited: Mar 15, 2008
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