I'm trying to find Fourier cosine series for function $$f(x) = \begin{cases} \frac{\pi}{2}-x,\quad &\text{if } x\in\left[0,\frac{\pi}{2}\right) \\ \pi,\quad &\text{if } x\in\left[\frac{\pi}{2},\pi\right] \end{cases}$$ in $[0,\pi]$.
I found coefficients $$b_k=0,\quad a_0=\frac{5\pi}{4}\quad \text{and}\quad a_k=\frac{2}{\pi}\cdot\frac{1-\cos\frac{k\pi}{2} + k\pi\sin k\pi - k\pi\sin\frac{k\pi}{2}}{k^2}$$ where $k\in\mathbb{N}$, but I don't know how to simplify $a_k$ to write cosine series as $$\frac{5\pi}{4}+\sum_{k=1}^{\infty}\ ...\ \cos kx$$ and investigate pointwise convergence in $[0,\pi]$.
Can anyone help me with this?