Zeta(2) and Pi Squared Over Six
The sum of the reciprocals of the squares of the positive integers
was discovered by the Swiss mathematical genius Leonahard
Euler in 1734. This series today is a
special case of zeta function when s = 2, namely, 
.
Euler combined the knowledge of
trigonometry and calculus to derive (A) as follows:
Using calculus to derive MacLaurin series for the function f(x) = sin(x):
Let
be
a polynomial with power n, its roots are 
. Therefore, P(x) can be rewritten in the form
as follows:
From (2), we notice that at x = 0, P(0) = 1, and 
. We can express (2) into (1) by multiplying
the terms and collecting like terms.
Now, use the infinite product
We notice that when x approaches to 0, both sides of (3) are
equal to 1. It means that the roots of
equation (3) satisfy the properties of the equation (2). Hence, multiplying the terms and collecting
like terms. Finally, comparing
coefficients of
in
Eqn. (1) and Eqn. (3) gives
.
Using a similar method, Euler continued to derive other sum of reciprocals of even powers of 4, 6, 8, 10, …
,
,
,
and
more
values of zeta function at even positive integers.
There are various methods to proof
series (A). One of them is to
use the sine series for f(x) = x, 0<x<
,
and the mean square value, obtain
Recall that the root mean square value of the function f(x) over an interval from x = a to x = b is defined as:
The interval is of length
,
hence,
On the other hand, for f(x) expressed in its Fourier series as
,
then
Integrations from s to s+
of
the last two terms vanish. The other
terms vanish due to its symmetry about the x-axis and orthogonal property, m =
n+1.
.
The values of
and
,
n = 0, 1, 2, 3, …; therefore, we obtain
.
Recall that f(x) = x, where 0<x<
,
hence
Equating the expressions (4) and (5), we obtain
.