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Have a look at https://en.wikipedia.org/wiki/Basel_problem https://en.wikipedia.org/wiki/Basel_problem . I'm not a mathematician, and I'm probably wrong, but I
by Intermernet 11y ago
Have a look at https://en.wikipedia.org/wiki/Basel_problem https://en.wikipedia.org/wiki/Basel_problem . I'm not a mathematician, and I'm probably wrong, but I think it has to do with infinite series that "circle" around a value, and asymptotically approach a value, only accurately expressed in terms of pi.
"Euler's original derivation of the value π2/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series."
Sorry, not very convincing or geometric, and I'm sure someone else can provide a better answer, but that's how I visualize it.
- impendia 11y agoNo apologies necessary, I think that's just the state of mathematics. The sum of the inverse fourth powers is pi^4/90. Sixth powers, pi^6/915. The pattern keeps going for even powers and not for odd. I don't think there is a geometric proof. But if there were, it would be incredible.
- darkmighty 11y agoA simple explanation I enjoy (not quite visual, but geometric in some sense) is the on the wiki using Fourier series, understanding it requires just a few steps: 1) You accept Parseval's identity: by changing between orthogonal basis, the energy (sum of absolute squared values) remains the same. 2) You find that the energy of the periodic function x mod pi is sum(1/n^2). 3) Evaluating the integral gives pi^2/6! The square is neatly explained because it involves energies. I guess this is why the pattern works only for even powers.