Evaluating $\sum_{n=1}^{\infty} \int_{1}^{\infty} \frac {\cos(2 \pi n t)}{t^z} dt$ I need help evaluating any of:
$$\sum_{n=1}^{\infty} \int_{1}^{\infty} \frac {\cos(2 \pi n t)}{t^z} dt$$
$$\sum_{n=1}^{\infty} \int_{1}^{\infty} \frac {\cos(2 \pi (2n - 1) t)}{t^z} dt$$
Where $z \in \Bbb{C}$

Also I would like to ask if this is true:
$$\sum_{n=1}^{\infty} \int_{1}^{\infty} \frac {\cos(2 \pi n t)}{t^z} dt = \int_{1}^{\infty} \frac {\sum_{n=1}^{\infty} \cos(2 \pi n t)}{t^z} dt$$
If yes then how to in clever way calculate:
$$\sum_{n=1}^{\infty} \cos(2 \pi n t)$$

In a very general manner, the sum and integration operations can be switch iff they both converge in both cases.
For the evaluation of the integral I would do the following:
$$
\sum_{n=1}^\infty \int_0^\infty \frac{\cos(2\pi nt)}{t^z}dt=\sum_{n=1}^\infty \int_0^\infty \frac{\frac{e^{i2\pi nt}-e^{-i2\pi nt}}{2}}{t^z}dt\\
\frac{1}{2}\sum_{n=1}^\infty \int_0^\infty \frac{e^{i2\pi nt}-e^{-i2\pi nt}}{t^z}dt
$$
Now try interchanging the sum and integral, I think this would be a good way to approach the problem
