Why does the sum of the volumes of all even dimensional n-spheres with radius 1 converge to e^pi?


I can understand high school level math but not much more. Thx in advance for your reply!

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2 Answers

Anonymous 0 Comments

Just to be clear, you say you can only understand high school math, and then proceed to ask about algebraic topology, a masters level math course?

Anonymous 0 Comments

This isn’t ELI5, but it’s an explanation, and I’m sure someone with more knowledge in the subject will be able to possibly simplify this down a little bit.

Firstly, we need to find the formula of the volume for an n-ball. The proof for the formula can be found [here](https://en.m.wikipedia.org/wiki/Volume_of_an_n-ball#proofs), however, the methods involve some form of integration (you won’t see this until calculus, and most of the proofs use iterated integrals, which you won’t see until Calc III), or a geometric proof, which uses arguably more complicated math in it’s process.

Anyways, at the end of that process, we end up with a formula for the volume of an n-ball. Yay! With a little bit more work (which might even be possible at the high school level), we can find the formula for the volume of an even dimensional n-ball with radius R. Since we are are only concerned with R = 1, we have V = π^(k)/k! .

Now we get to the plug and chug part of the problem. Since we are summing all of the even dimensional volumes, we have the sum from k = 0 to infinity of π^(k)/k! . You might not know this at the high school level, but it’s very common to actually define exp(x) (which is equal to e^x) as an infinite degree polynomial. And it just so happens that this definition makes things incredibly easy as e^x = the sum from k = 0 to infinity of x^(k)/k! . Notice the left hand is exactly what we want to evaluate with x = π, and hence our answer is e^(π).

I hope this helps a little bit, even though a lot of the math is quite involved.