Euler's summmation of 1/n^2Date: 03/15/2000 at 23:51:55 From: Brad Bennett Subject: Euler's summmation of 1/n^2 How do you go about proving that pi^2/6 = the summation of 1/n^2 from 1 to infinity? What mathematical method could I use to solve the problem? Date: 03/16/2000 at 16:38:46 From: Doctor Schwa Subject: Re: Euler's summmation of 1/n^2 This is a hard problem! A good discussion can be found in George Polya's excellent book, _Mathematics and Plausible Reasoning_. A short answer: Consider the function sin (pi*x). (I know you're asking "what does sin (pi x) have to do with the sum of 1/n^2?" but bear with me). On the one hand, it's a function that's zero for every integer k. So, as an infinite polynomial product, it must be true that sin (pi x) = a x (1-x) (1+x) (1-x/2) (1+x/2) (1-x/3) (1+x/3) ... so that the right-hand side has zeros in all the right places. By difference of squares, we get sin (pi x) = a x (1-x^2) (1-x^2/4) (1-x^2/9) ... Now the left side can also be expanded in a power series, sin (pi x) = (pi x) - (pi x)^3 / 3! + ... Using that, we first discover that a = pi to make the x coefficients match up, and then discover that the sum of 1/n^2 is pi^2/6 to make the x^3 coefficients match up. Feel free to write back with questions about all the steps I skipped in this derivation. It's a really neat problem! - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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