Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

One method is to consider the generating function of $\zeta(2k)$: $$ \begin{align} f(x) &=\sum_{k=1}^\infty\zeta(2k)\,x^{2k}\\ &=\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{x^{2k}}{n^{2k}}\\ &=\sum_{n=1}^\infty\frac{x^2/n^2}{1-x^2/n^2}\\ &=\sum_{n=1}^\infty\frac{x^2}{n^2-x^2}\\ &=-\frac{x}{2}\sum_{n=1}^\infty\left(\frac{1}{x-n}+\frac{1}{x+n}\right)\\ &=-\frac{x}{2}\left(\pi\cot(\pi x)-\frac1x\right)\\ &=\frac12(1-\pi x\cot(\pi x))\tag{1} \end{align} $$ In light of equation $(1)$, find the power series of $$ x\cot(x)=\sum_{k=0}^\infty a_kx^{2k} $$ $$ \cos(x)=\frac{\sin(x)}{x}\sum_{k=0}^\infty a_kx^{2k} $$ $$ \begin{align} \sum_{n=0}^\infty(-1)^n\!\frac{x^{2n}}{(2n)!} &=\sum_{n=0}^\infty(-1)^n\!\frac{x^{2n}}{(2n+1)!}\;\;\sum_{k=0}^\infty a_kx^{2k}\\ &=\sum_{n=0}^\infty\left(\sum_{k=0}^n(-1)^k\!\frac{a_{n-k}}{(2k+1)!}\right)x^{2n}\tag{2} \end{align} $$ Comparing the coefficients of the powers of $x$ in $(2)$ yields $$ \begin{align} a_n &=\frac{(-1)^n}{(2n)!}-\sum_{k=1}^n(-1)^k\!\frac{a_{n-k}}{(2k+1)!}\\ &=\frac{(-1)^n2n}{(2n+1)!}-\sum_{k=1}^{n-1}(-1)^k\!\frac{a_{n-k}}{(2k+1)!}\tag{3} \end{align} $$ Since $a_n=-2\dfrac{\zeta(2n)}{\pi^{2n}}$ for positive $n$, $(3)$ becomes $$ \zeta(2n)=\frac{(-1)^{n-1}\pi^{2n}}{(2n+1)!}n+\sum_{k=1}^{n-1}\!\frac{(-1)^{k-1}\pi^{2k}}{(2k+1)!}\zeta(2n-2k)\tag{4} $$ Equation $(4)$ gives $\zeta(2n)$ recursively for positive $n$: $$ \begin{align} \zeta(2)&=\frac{\pi^2}{3!}=\frac{\pi^2}{6}\\ \zeta(4)&=-\frac{\pi^4}{5!}2+\frac{\pi^2}{3!}\zeta(2)=\frac{\pi^4}{90}\\ \zeta(6)&=\frac{\pi^6}{7!}3-\frac{\pi^4}{5!}\zeta(2)+\frac{\pi^2}{3!}\zeta(4)=\frac{\pi^6}{945}\\ \zeta(8)&=-\frac{\pi^8}{9!}4+\frac{\pi^6}{7!}\zeta(2)-\frac{\pi^4}{5!}\zeta(4)+\frac{\pi^2}{3!}\zeta(6)=\frac{\pi^8}{9450} \end{align} $$

I have posted here in Portuguese a recursive method based on the computation of the Fourier trigonometric series expansion for the function defined in $\left[ -\pi ,\pi \right] $ by $f(x)=x^{2p}$ and extended to all of ${\mathbb R}$ periodically with period $2\pi.$ This is a shorter description than the original. In this reply I outline the case $\zeta(4)$. For $p=3$ the expansion is

$$x^{6}=\dfrac{\pi ^{6}}{7}+2\displaystyle\sum_{n\ge 1}^{}\left( \left( \dfrac{6}{n^{2}}\pi ^{4}-\dfrac{120}{n^{4}}\pi ^{2}+\dfrac{720 }{n^{6}}\right)\cos n\pi \right) \cos nx.\tag{1}$$

The computation is as follows:

$$\begin{equation*} f(x)=x^{2p}=\frac{a_{0,2p}}{2}+\sum_{n=1}^{\infty }\left( a_{n,2p}\cos nx+b_{n,2p}\sin nx\right) , \end{equation*}$$ where the coefficients are given by the following integrals $$\begin{eqnarray*} a_{0,2p} &=&\frac{1}{\pi }\int_{-\pi }^{\pi }x^{2p}\;\mathrm{d}x=\frac{2\pi ^{2p}}{2p+1}, \\ a_{n,2p} &=&\frac{1}{\pi }\int_{-\pi }^{\pi }x^{2p}\cos nx\;\mathrm{d}x=\frac{2}{\pi } \int_{0}^{\pi }x^{2p}\cos nx\;\mathrm{d}x, \\ b_{n,2p} &=&\frac{1}{\pi }\int_{-\pi }^{\pi }x^{2p}\sin nx\;\mathrm{d}x=0. \end{eqnarray*}$$ The series expansion is thus $$\begin{equation*} x^{2p}=\frac{\pi ^{2p}}{2p+1}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos nx\int_{0}^{\pi }x^{2p}\cos nx\;\mathrm{d}x.\tag{2} \end{equation*}$$ For $f(\pi )=\pi ^{2p}$ we obtain $$ \begin{equation*} \pi ^{2p}=\frac{\pi ^{2p}}{2p+1}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi \int_{0}^{\pi }x^{2p}\cos nx\;\mathrm{d}x, \end{equation*}$$ where the integral $$ \begin{equation*} I_{n,2p}:=\int_{0}^{\pi }x^{2p}\cos nx\;\mathrm{d}x \end{equation*}$$ satisfies the following recurrence, as can be shown by integration by parts $$\begin{equation*} I_{n,2p}=\frac{2p}{n^{2}}\pi ^{2p-1}\cos n\pi -\frac{2p(2p-1)}{n^{2}} I_{n,2\left( p-1\right) },\qquad I_{n,0}=0.\tag{3} \end{equation*}$$

• For $p=1$, we get $$\begin{equation*} I_{n,2}=\frac{2}{n^{2}}\pi\cos n\pi. \end{equation*}$$ and $$\begin{eqnarray*} \pi ^{2} &=&\frac{\pi ^{2}}{3}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi \cdot I_{n,2} \\ &=&\frac{\pi ^{2}}{3}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi \left( \frac{2}{n^{2}}\pi \cos n\pi \right) \\ &=&\frac{\pi ^{2}}{3}+4\sum_{n=1}^{\infty }\frac{1}{n^{2}} \\ &\Rightarrow &\zeta (2)=\sum_{n=1}^{\infty }\frac{1}{n^{2}}=\frac{\pi ^{2}}{6 } \end{eqnarray*}$$
• For $p=2$, we get $$ \begin{equation*} I_{n,4}=\left( \frac{4\pi ^{3}}{n^{2}}-\frac{24\pi }{n^{4}}\right) \cos n\pi \end{equation*}$$ and $$ \begin{eqnarray*} \pi ^{4} &=&\frac{\pi ^{4}}{5}+\frac{2}{\pi }\sum_{n=1}^{\infty }\cos n\pi \cdot I_{n,4}=\frac{\pi ^{4}}{5}+\frac{4\pi ^{4}}{3}-48\sum_{n=1}^{\infty } \frac{1}{n^{4}} \\ &\Rightarrow &\zeta (4)=\sum_{n=1}^{\infty }\frac{1}{n^{4}}=\frac{\pi ^{4}}{ 90}. \end{eqnarray*}$$
• Finally for $p=3$, we get $$\begin{equation*} I_{n,6}=\left( \frac{6\pi ^{5}}{n^{2}}-\frac{120\pi ^{3}}{n^{4}}+\frac{720}{ n^{6}}\right) \cos n\pi \end{equation*}$$ and $$ \begin{equation*} \pi ^{6}=\frac{\pi ^{6}}{7}+2\sum_{n=1}^{\infty }\left( \frac{6\pi ^{4}}{ n^{2}}-\frac{120\pi ^{2}}{n^{4}}+\frac{720}{n^{6}}\right), \end{equation*}$$ from which the result follows $$\zeta(6)= \begin{equation*} \sum_{n=1}^{\infty }\frac{1}{n^{6}}=\frac{\pi ^{6}}{945}. \end{equation*}$$

Plots of the periodic function defined in $\left[ -\pi ,\pi \right] $ by $f(x)=x^{6}$ (blue curve) and of the partial sum with the first 10 terms of its Fourier trigonometric series (red curve).

This method generates recursively the sequence $(\zeta(2p))_{p\ge 1}$.

You may start with (for $x \in [0,1]$)
$$B_1(x)=x-\frac 12=2\sum_{n=1}^{\infty} \frac{(-1)^n}{2\pi n}\sin\left(2\pi n\left(x-\frac 12\right)\right)$$

and integrate multiple times $x-\frac 12$ (adding the necessary constant when required!) to get :

$$B_{2k}(x)=2(-1)^k\sum_{n=1}^{\infty} \frac{(-1)^n}{(2\pi n)^{2k}}\cos\left(2\pi n\left(x-\frac 12\right)\right)$$ (this is detailed in this paper page 6)

You could too use following formula $$ \cot(\pi z)=\frac1{\pi}\left[\frac1{z}-\sum_{k=1}^{\infty}\frac{2z}{k^2-z^2}\right] $$ (obtained from computation of Fourier series of $\cos(zx)$ for $-\pi \le x \le \pi$ with the result :

$$ \cos(zx)=\frac{2z\sin(\pi z)}{\pi}\left[\frac1{2z^2}+\frac{\cos(1x)}{1^2-z^2}-\frac{\cos(2x)}{2^2-z^2}+\frac{\cos(3x)}{3^2-z^2}-\cdots\right] $$ applied to $x=\pi$)

and deduce an interesting expansion of $\frac z2\left(\cot\left(\frac z2\right)\right)$ in powers of $z^2$.