using periodicity of a funtion

Hello!

I got the following function:

The function has a periodicity of .

Let be the binary expansion of a number between 0 and 1.

Letīs consider .

If we look at

why do we have by the periodicity of for i > n

?

I guess that the period is subtracted exactly i times, but I can not show it.

Can anyone help please? Thanks.

Re: using periodicity of a funtion

For , we have and you have to show that for an integer .

Re: using periodicity of a funtion

Quote:

Originally Posted by

**girdav** For

, we have

and you have to show that

for an integer

.

Your advice has been great! :)

What is left to show is not equivalent to show that

is an integer, correct?. I guess I actually have to calculate that integer j?

Thanks.

Re: using periodicity of a funtion

You have to show that is an integer. Use the fact that

Re: using periodicity of a funtion

Quote:

Originally Posted by

**girdav** You have to show that

is an integer. Use the fact that

Yeah! That is what I did. I just wasnīt sure I had to show exactly that one cause I first thought I had to calculate a specific j.

I know that and then . Thus the sum will be an integer.