# 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

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