I have the following function: \begin{align*} f(x) = \begin{cases} 0, & \text{if } 0 \leq x < \frac{1}{5} \\ 5x-1, & \text{if } \frac{1}{5} \leq x < \frac{2}{5} \\ 1, & \text{if } \frac{2}{5} \leq x < \frac{3}{5} \\ -5x+4, & \text{if } \frac{3}{5} \leq x < \frac{4}{5} \\ 0, & \text{if } \frac{4}{5} \leq x \leq 1 \end{cases} \end{align*}
and define $ a_{0} = \int_{0}^{1} \! f(x) \, dx , a_{k}= 2 \int_{0}^{1} \! f(x) \cos(2kx\pi)\, dx, , b_{k}= 2 \int_{0}^{1} \! f(t)\sin(2kx\pi)\, dx $
Consider the series of functions in [0, 1] whose sequence of partial sums $\{F_n\}$ is defined by
\begin{equation} F_{n}= a_{0} + \displaystyle\sum_{k=1}^{n}(a_{k}cos(2kx\pi) + b_{k}\sin(2kx\pi)) \end{equation}
Formally prove that $F_{n} \rightarrow f$ punctually in $[0,1]$.
I calculate the values of $a_{0}$, $a_{k}$ and $b_{k}$ which are given by \begin{align*} a_{0} &= \int_{0}^{1} f(t) ,dt \\ &= \frac{4}{10} \end{align*} \begin{align*} a_{k} &= 2\int_{0}^{1} f(x) \cos(2kx\pi) dx \\ &=\frac{5}{2{\pi}^2}\left[ \dfrac{\cos\left(\frac{4{\pi}k}{5}\right)-\cos\left(\frac{2{\pi}k}{5}\right)-\cos\left(\frac{8{\pi}k}{5}\right)+\cos\left(\frac{6{\pi}k}{5}\right)}{k^2}\right] \end{align*}
\begin{align*} b_{k} &= 2\int_{0}^{1} f(x) \sin(2kx\pi) dx \\ &=\frac{5}{2{\pi}^2}\left[ \dfrac{\sin\left(\frac{4{\pi}k}{5}\right)-\sin\left(\frac{2{\pi}k}{5}\right)-\sin\left(\frac{8{\pi}k}{5}\right)+\sin\left(\frac{6{\pi}k}{5}\right)}{k^2}\right] \ \end{align*}
my idea is to use dirichlet's theorem for Fourier series, for which I make a substitution of the form S=$2\pi x-\pi$ since the theorem states that it must be piecewise continuous in $[-\pi,\pi]$ but then making this substitution I am left with $a_{k}$ and $b_{k}$
$a_{k}= 2 \int_{-\pi}^{\pi} \! f(\frac{s+\pi}{2\pi}) \cos(ks)\, ds$
$b_{k}= 2 \int_{-\pi}^{\pi} \! f(\frac{s+\pi}{2\pi}) \sin(ks)\, ds$
I already have all the hypotheses of the theorem, but I don't know how to conclude to say that the function converges point-wise obviously also changing $a_{0}$
Is this correct? what can I do better? I appreciate any help!