Euler's Faith and FollyDate: 03/25/2011 at 12:43:07 From: sankar Subject: Eulers infinite sums Hi, I am reading a book titled "Journey through Genius," by William Dunham. There is one chapter in this book about how Euler arrived at the sum of the reciprocals of the squares of the whole numbers, which is pi^2/6. The way he proves this is by using the Taylor expansion of sin(x)/x and equating the expansion with factors of the equation sin(x)/x. Then he collects all the coefficients of x^2 on right side with x^2 coefficient of the left side. But at the end of the chapter, Dunham says this: "Today, we recognize that Euler was not so precise in his use of the infinite as he should have been. His belief that finitely generated patterns and formulas automatically extend to the infinite case was more a matter of faith than of science, and subsequent mathematicians would provide scores of examples showing the folly of such hasty generalizations." I don't understand what is wrong with Euler's analysis. Date: 03/25/2011 at 19:24:58 From: Doctor Jordan Subject: Re: Eulers infinite sums Hi Sankar, Let's say we have the polynomial f(x) = x^3 + x This can be written as f(x) = x(x^2 + 1) Before the use of complex numbers, we would have said that the only root of this polynomial is 0. But then it would be false that f(x) is a product of linear factors x - a, where a is a root of f(x). If we use complex numbers, then the roots of f(x) = x^3 + x are 0, i and -i; and indeed, f(x) = x(x - i)(x + i) So to be able to factor a polynomial, we need to know all of its complex roots. If we want to write sin(x) as a product of linear factors x - a, where a is a root of sin(x), then -- in analogy to factoring a polynomial -- we want to know all of the complex factors of sin(x). A criticism Euler received from, I believe, Nicolaus Bernoulli claims that determining the integer multiples of pi as the *real* roots of sin(x) does not mean that these are the function's *only* roots. In the same way that f(x) = x^3 + x has the real root 0, but also the complex roots i and -i, perhaps there are other complex roots. One could also complain that even if we know that the only roots of sin(x) are the integer multiples of pi, how do we know that sin(x) is equal to the infinite product (x - a), where a is a root of sin(x)? Using his formula e^(ix) = cos(x) + isin(x), Euler later did show that the only roots of sin(x) are the integer multiples of pi. And the equivalence of sin(x) and this infinite product can be shown in several different ways. The argument that uses the least prior knowledge that I know of is given on p. 18 of Reinhold Remmert's "Classical Topics in Complex Function Theory," a page available on Google Books. - Doctor Jordan, The Math Forum http://mathforum.org/dr.math/ Date: 03/27/2011 at 01:13:03 From: sankar Subject: Thank you (Eulers infinite sums) Thank you so much. This really enhanced my understanding of the argument. I really want to appreciate the work you guys are doing. |
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