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Proving that 'volume' and 'surface' of hypersphere go to 0 as n -> infinity?

  1. Mar 7, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm supposed to find the equations of a hypersphere in n-dimensions (meaning the set of points within the radius R), as well as of its surface (the set of points at exactly radius R). I've already found the equations, and now need to show that both go to zero as n goes to infinity.

    2. Relevant equations

    [itex]V_n(R) = \frac{\pi^{n/2} R^n}{\Gamma(n/2+1)}[/itex]
    [itex]S_n(R) = \frac{2 \pi^{n/2} R^{n-1}}{\Gamma(n/2)}[/itex]

    3. The attempt at a solution
    I know that both go to zero just by observation, but that's not really mathematical. I was able to show that, if I put the hypersphere within a hypercube of side length A and subtract their volumes, it goes to An, implying that the difference in their volumes goes to the volume of the hypercube. But I'm not sure how solid that is -- I feel it's more of an argument for the volume going to zero rather than a proof. And it doesn't help me with the surface 'area' anyway.

    EDIT: I also considered using Stirling's approximation:
    [itex] \Gamma(n+1) \approx \left( \frac{n}{e} \right)^n \sqrt{2 \pi n} [/itex]

    Then, inputting that into the above for Vn, I get:

    [itex]\lim_{n \to \infty} \frac{\pi^{n/2} e^{n} 2^{n} R^n}{n^n} [/itex]

    I suppose that's a decent way of showing that the limit is equal to zero?
     
    Last edited: Mar 7, 2012
  2. jcsd
  3. Mar 7, 2012 #2
    The denominators you have are gamma functions, they grow faster than exponentials, I think you can just claim that S and V go to zero, the details are probably not very interesting anyway.
     
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