I want to calculate $$\sum_{k=-\infty}^{\infty}\frac{e^{ika}}{1+|k|^r}$$ for $r\geq 2$ and $r\in N$. \begin{align*}\sum_{k=-\infty}^{\infty}\frac{e^{ika}}{1+|k|^r}&=1+\sum_{k=1}^{\infty}\frac{e^{ik\lambda}}{1+k^r}+\sum_{k=-\infty}^{-1}\frac{e^{ika}}{1+|k|^r}\\ &=1+\sum_{k=1}^{\infty}\frac{e^{ika}}{1+k^r}+\sum_{k=1}^{\infty}\frac{e^{-ika}}{1+k^r}\\ &=1+2\sum_{k=1}^{\infty}\frac{\cos(ka)}{1+k^r} \end{align*}
I don't know how to calculate $\sum_{k=1}^{\infty}\frac{\cos(ka)}{1+k^r}$.
Any help, even for a specific $r$, is appreciated.