I have trouble solving this sum. I already rewrite it, but I don't know what to do next, also if it is even possible to solve it ?
Let:
$z = a + x i$
$\sum_{n=1}^{\infty} (E_{z}(i \pi n) + E_{z}(- i \pi n))$
Then:
$ \sum_{n=1}^{\infty} (E_{z}(i \pi n) + E_{z}(- i \pi n)) = \sum_{n=1}^{\infty} ( \int_{1}^{\infty} e^{- i \pi n t} t^{-z}dt + \int_{1}^{\infty} e^{i \pi n t} t^{-z}dt ) = \sum_{n=1}^{\infty} (\int_{1}^{\infty} \frac{2 cos(\pi n t)}{t^z} dt) = 2 \sum_{n=1}^{\infty} (\int_{1}^{\infty} \frac{cos(\pi n t)}{t^z} dt) $
And also can I write it as:
$ 2 \int_1^\infty \frac{\sum_{n=1}^\infty cos(\pi n t)}{t^z} dt $
If not, then why ?