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):
Letbe 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 ofin 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, …
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
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
, 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