# How to calculate this sum?

• Zhivago
In summary, the conversation discusses the series \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. The participants share different sources where this series is mentioned, including a paper, a book, and a Google link. They also mention that the book provides a clear proof for the series. Overall, the participants find the series and its properties fascinating.

#### 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)

That's a pretty amazing series. I found some more information about it in the following paper. See : http://www.jstor.org/view/00029890/di991774/99p0626c/0

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Curses, JSTOR!

uart said:
That's a pretty amazing series. I found some more information about it in the following paper. See : http://www.jstor.org/view/00029890/di991774/99p0626c/0

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.

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Zhivago said:
...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.

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!

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## 1. How do I calculate the sum of two numbers?

To calculate the sum of two numbers, you simply add them together. For example, to find the sum of 5 and 7, you would add 5 + 7 = 12. This is the basic method for calculating sums.

## 2. What is the order of operations when calculating a sum?

The order of operations when calculating a sum is known as PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This means that you should first solve any operations within parentheses, then solve any exponents, followed by multiplication and division from left to right, and finally addition and subtraction from left to right.

## 3. How do I calculate the sum of more than two numbers?

To calculate the sum of more than two numbers, you can use the same method as before. Simply add all the numbers together. For example, to find the sum of 3, 6, and 9, you would add 3 + 6 + 9 = 18. You can also use a calculator for more complex sums with multiple numbers.

## 4. What is the difference between a sum and a total?

A sum is the result of adding two or more numbers together, while a total refers to the final amount or final sum of a calculation. For example, if you add 5 and 7, the sum is 12, but the total amount would depend on what you are adding the numbers for.

## 5. How do I know if I have calculated a sum correctly?

To check if you have calculated a sum correctly, you can use the inverse operation. For example, if you added two numbers, you can subtract one number from the sum and see if it equals the other number. Additionally, you can use a calculator to double check your work.