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Euler's Faith and Folly

Date: 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.
Associated Topics:
High School History/Biography
High School Imaginary/Complex Numbers
High School Polynomials
High School Sequences, Series

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