Hi all,

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?

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- May 13th 2011, 03:34 AMmeasuremanIntegration over R^n
Hi all,

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? - May 13th 2011, 06:21 AMOpalg
If you think in terms of computing the integral using a spherical shell method, you get

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 - May 13th 2011, 08:28 AMmeasureman
- May 13th 2011, 10:09 AMdavismj
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.