calculate the Fourier series of the $2\pi$-periodic continuation of
$$f(t):=|t|, \quad t\in[-\pi,\pi)\tag{1}$$
We know that
$$f(t)=\sum_{k=-N}^N c_k\cdot e^{ikt}\quad \&\quad c_k=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-ikt}dt\tag{2}$$
So let's calcualte the $c_k$.
$$c_k=\frac{1}{2\pi}\int_{-pi}^\pi|t|\cdot e^{-ikt}dt=\frac{2}{2\pi}\int_0^\pi t\cdot \cos(tk) dt=\frac{1}{\pi}[t\cdot\sin(tk)]_0^\pi+\frac{1}{\pi}[\cos(tk)]_0^\pi=0\frac{1}{\pi}(-1-1)=\frac{-2}{\pi}\tag{3}$$
Whereas we used the identity $\int_{-a}^a|t|e^{-ikt}dt=2\int_0^a t\cos(kt)dt$ Explanation (wheres the sub $t\to -t$ was used in the 2nd step)
So we get
$$f(t)=\sum_{k=-\infty}^\infty \frac{-2}{\pi}e^{ikt}\tag{4}$$
Sadly I don't have any solutions. Is that correct?