How do you find #\sum _ { n = 1} ^ { \infty } ( \frac { 1} { n ^ { 2} } ) = \frac { pi ^ { 2} } { 6}#?

1 Answer
George C.
Nov 23, 2017

Consider:

#sin(x)/x = prod_(n=1)^oo (1-x/(npi))(1+x/(npi))#

and look at the coefficient of #x^2#

Explanation:

Finding the sum #sum_(n=1)^oo 1/n^2# is the Basel problem, first posed in 1644 by Pietro Mengoli and solved by Leonhard Euler in 1734.

See https://socratic.org/s/aL67DCh9 for my favourite method of solving it, which uses the Weierstrass Factorisation Theorem and:

#sin(x)/x = prod_(n=1)^oo (1-x/(npi))(1+x/(npi))#

by looking at the coefficient of #x^2#

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