Triangle Wave Fourier Series

Calculus Level 2

Find the Fourier series of the triangle wave which is defined by

$f(x) = -2|x-0.5|+1$

for $-0.5\leq x \leq 1.5$ and is periodic outside this region.

Hint: Try plotting the given function first.

\frac{8}{\pi^2} \sum_{k \text{ odd}} \frac{(-1)^{(k-1)}}{k} \sin \pi k x

\frac{8}{\pi^2} \sum_{k \text{ even}} \frac{(-1)^{k/2}}{k} \sin \pi k x

\frac{8}{\pi^2} \sum_{k \text{ odd}} \frac{(-1)^{(k-1)/2}}{k^2} \sin \pi k x

\frac{8}{\pi^2} \sum_{k \text{ odd}} \frac{(-1)^{k}}{k} \sin 2 \pi k x