How to calculate this sum?

1. Mar 5, 2008

Zhivago

$$\sum_{n=1}^{\infty} \frac{o(2^n)}{2^n} = \frac{1}{9}$$
where $$o(2^n)$$ is the number of odd digits of $$2^n$$.

Found it in
http://mathworld.wolfram.com/DigitCount.html
equation (9)

2. Mar 10, 2008

uart

Last edited by a moderator: Apr 23, 2017
3. Mar 10, 2008

Tedjn

Curses, JSTOR!

4. Mar 14, 2008

Zhivago

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.

Last edited by a moderator: Apr 23, 2017
5. Mar 15, 2008

uart

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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