value of Dirichlet eta function at $s=2$

The value

$$\eta (2)=1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+-\mathrm{\dots }$$

of the Dirichlet eta function can be found by using the Fourier cosine series of the function  $x\mapsto x-x^2$ on the interval  $[0,\mathrm{\hspace{0.17em}1}]$:

$x-x^2={\displaystyle \frac{1}{6}}-{\displaystyle \frac{1}{{\pi }^2}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{\cos 2n\pi x}{n^2}}\mathit{\hspace{1em}}\text{for}\mathrm{\hspace{0.33em}\hspace{0.33em}0}\leqq x\leqq 1$ (1)

Substituting  $x:=\frac{1}{2}$  to the equation (1) yields

$$\frac{1}{4}=\frac{1}{6}-\frac{1}{{\pi }^2}\sum _{n=1}^{\mathrm{\infty }}\frac{\cos n\pi }{n^2}=\frac{1}{6}+\frac{1}{{\pi }^2}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n+1}}{n^2},$$

which we can solve to the form

$\eta (2)={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{n+1}}{n^2}}={\displaystyle \frac{{\pi }^2}{12}}.$ (2)

This result could be obtained very simply by using the functional equation connecting Dirichlet eta function to Riemann zeta function.

Combining the equation (2) with the result concerning the Riemann zeta function at 2 (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2) shows that

$1+{\displaystyle \frac{1}{3^2}}+{\displaystyle \frac{1}{5^2}}+{\displaystyle \frac{1}{7^2}}+\mathrm{\dots }={\displaystyle \frac{{\pi }^2}{8}}.$ (3)

Title	value of Dirichlet eta function at $s=2$
Canonical name	ValueOfDirichletEtaFunctionAtS2
Date of creation	2013-03-22 18:22:09
Last modified on	2013-03-22 18:22:09
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Result
Classification	msc 11M41
Related topic	CosineAtMultiplesOfStraightAngle
Related topic	ValueOfTheRiemannZetaFunctionAtS2