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)