Q4

[Victor Ivrii]

Decompose into full Fourier series on interval $[-\pi,\pi]$ and sketch the graph of the sum of such Fourier series:
\begin{equation}
f(x)=|x|.
\end{equation}

[Roro Sihui Yap]

Since $f(x) = |x|$ is an even function, $b_n = 0 \ \ \forall n$
$a_0 = \frac{1}{\pi}\int_{-\pi}^\pi |x|\,dx = \pi$
$a_n = \frac{1}{\pi}\int_{-\pi}^\pi |x|\cos(nx) \,dx = \int_0^\pi \frac{2x}{\pi}\cos(nx) \,dx $
Integrating by parts
$a_n =\frac{2x}{n\pi}\sin(nx) \big|_{0}^{\pi}- \int_0^\pi \frac{2}{n\pi} \sin(nx) \,dx  = \frac{2}{n^2\pi}\cos(nx)\big|_{0}^{\pi} $
$a_n = \begin{cases}\frac{-4}{n^2\pi} && n \ is \  odd \\0 && n \ is \  even\end{cases}$

$f(x) = \frac{\pi}{2} +\sum_{m=0}^\infty \frac{-4}{(2m+1)^2\pi}\cos((2m +1)x) $

[Victor Ivrii]

Everybody understands that on the graph bottoms are at $2n\pi $ and picks at $(2n+1)\pi$, $n\in \mathbb{Z}$.