I've been trying to solve a problem which has boiled down to whether the integral
exists for some large enough N, where .
I have been told that this does exist, but I don't know how I would prove this?
where is the "surface area" (or more correctly the (n–1)-dimensional volume) of the (n–1)-sphere. Since for some constant , it follows that and that converges for by comparison with the integral of
Yea, think of two-dimensions. Instead of integrating over in cartesian coordinates, you integrate first over a circle of radius r, and then you integrate r from 0 to infinity.
To understand why in n-dimensions we take the (n-1) dimensional volume or surface area, notice that when you integrate the circle of radius r, you're integrating in one dimension, over a line from theta = 0 to theta = 2pi. From there you pick up the last dimension by integrating the surface area for all radii.