Consider the function:
$ f(\theta) = \begin{cases} 0 & \text{if } |\theta| >\delta \\ 1-|\theta|/\delta & \text{if } |\theta| \leq \delta \end{cases} $
I need to show that:
$$f(\theta)= \frac{\delta}{2\pi}+ 2\sum _{n=1}^{\infty} \frac{1-cos(n \delta)}{n^2 \pi \delta} cos(n\theta)$$
but the thing is that this integral $\int_{-\delta}^{\delta}( 1-|\theta|/\delta )e^{-in \theta}d \theta$ is a litle bit complicated for me because of the absolute value and how to make the substitution to integrate by parts, and the other thing is that $\int_{-\delta}^{\delta}( 1-|\theta|/\delta )d\theta \not= \frac{\delta}{2\pi}$
I dont know what I am doing wrong, Can you help me to complete this please?. Thanks a lot.