Expressions with floors and ceilings ( and ) are usually troublesome to work with. There are cases where you can essentially remove them by a change of variables.
Turns out, one form that regularly comes up in my calculations is , and it bugged me a while before I figured out the right way of getting rid of them (sometimes).
Let us start by an abstract case:
where (for now), and where is some function that depends only on . grows by one each time doubles in magnitude. We can therefore write the preceding as
where you can verify that still progresses from to as before, but we now have a different grouping what allows us to replace by , and factor it out. We can now simplify the sums depending on as a function of as
where is an expression depending only on . Finally, you can obtain a simplified expression for , the initial goal.
So, where does the bounds come from? As I said earlier, has natural boundaries of the form , with . Indeed:
The generator of the series at the far right of the table isn’t very hard to guess: .
It is a special case of a geometric series,
With , , it does simplify to the expected:
So by knowing how to sum powers of two (or any other constant, for that matter), we could remove and get a simpler expression. While the change of variable and adapting the sums’ boundaries helps a great deal, it is the fact that the base of the logarithm is an integer that helps quite a lot. Indeed, can you see what’s the problem is the base of the logarithm is now ?
You guessed it right. Now, instead than having boundaries at integers of the form , we must find where the function changes between some and , which can be quite messier.