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

In summary, the equations for a hypersphere in n-dimensions and its surface have been found, and it is required to show that both equations approach zero as n goes to infinity. One approach is to use Stirling's approximation and show that the limit is equal to zero, while another is to observe that the difference in volumes between the hypersphere and a hypercube approaches the volume of the hypercube. However, the details of the proof may not be necessary.
  • #1
hb1547
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Homework Statement


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.

Homework 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]

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?
 
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  • #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.
 

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

1. What is a hypersphere?

A hypersphere is a higher-dimensional analogue of a sphere in three-dimensional space. It is defined as the set of all points in n-dimensional space that are a fixed distance from a central point, where n is the number of dimensions. In three dimensions, a hypersphere is equivalent to a traditional sphere.

2. What is meant by 'volume' and 'surface' of a hypersphere?

The volume of a hypersphere refers to the amount of space enclosed by its surface, while the surface of a hypersphere refers to the boundary that separates the inside from the outside. In simpler terms, the volume is the 3-dimensional equivalent of area, and the surface is the 3-dimensional equivalent of perimeter.

3. How does the 'volume' and 'surface' of a hypersphere change as n increases?

As n increases, the 'volume' and 'surface' of a hypersphere both decrease. This is because as the number of dimensions increases, the distance between points on the surface of the hypersphere increases, making the surface area and volume of the hypersphere smaller.

4. Why is it important to prove that the 'volume' and 'surface' of a hypersphere go to 0 as n -> infinity?

This proof is important in mathematics because it helps us understand the behavior of higher-dimensional objects. It also has applications in physics and engineering, where understanding the properties of higher-dimensional objects is crucial in solving complex problems.

5. How is the proof of the 'volume' and 'surface' of a hypersphere going to 0 as n -> infinity carried out?

The proof involves using mathematical concepts such as integration and limits to calculate the volume and surface of a hypersphere in n dimensions. As n approaches infinity, the calculations will show that both the volume and surface tend towards 0, providing a mathematical proof of the statement.

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