How we're going to characterize that.

And that's a really interesting function to take a look at.

What I'm going to try to do with this analysis is try to

isolate the periodic terms.

There's an oscillation in here and

I want to try to isolate the part that oscillates.

The way that's going to happen is, we're going to take the function,

log base 2 of N, and we're going to work with the integer part of log base 2 of N.

And then the fluctuation is every time you come to an integer,

as you go from one integer to the next, there's fluctuation in the function.

Where as you're working with the real function log x, but

you're only picking off the integer parts of it.

So let's see how it works.

So all we're going to do is take that infinite sum and

we're going to break it into two parts.

The part where j is less than floor of log N, and

the part where it's greater than or equal to floor of log N.

So just split the sum into two parts at that one point.

And the key thing to notice about this is when j gets bigger than log base 2 of N,

then 2 to the j is going to be bigger than N.

And so we're going to have -N over something huge.

We're going to get a number very close to 1.

It's going to go away.

When j is small, like say j is 2 or something,

we have e to the -N, which is tiny.

The sum is just one.

So basically, were going to have log N terms that are very close to 1 and

all the rest of them are going to be very close to 0,

with just a little bit left in the center.

So let's look at how that goes.

In this case, we just split off the log N so we get floor of log N.

And then we have the second one, e to the -N / 2 to the j,

and that's just splitting the first sum into two parts.

And then this one is the same, so no change there.