For any number , we can express the difference between and raised to some power as:
To find which number most exceeds , we can find the maximum point of the above function.
First the derivative:
And then the maximum (assuming that the power, , is greater than 1):
We can relabel this final equation as a function in and which will except a power and yield the number which most exceeds itself raised to that power:
This function can be shown to converge to :
This is equivalent to saying that the number which most exceeds is ( most exceeds ). Remember that (because it gets weirder!).
Now the function describing the difference between these "greatest" numbers and their respective powers is:
This can also be shown to converge to :
This is equivalent to saying that the difference between the number which most exceeds () and () is .
This is understandably counter-intuitive, but taken in this context, it seems provable!
To finish, here is the grand consequence of this discussion: