$$\frac1{\operatorname{ch}(\frac\pi2)}-\frac1{3\operatorname{ch}(\frac{3\pi}2)}+\frac1{5\operatorname{ch}(\frac{5\pi}2)}-\cdots$$
How can I sum the above series in the image with the residue theorem? I solved the first two but failed with the third for that $zf(z)$ does not converge to 0 at $|z|\to\infty$, making the hints(methods) in the second image seem unavailable.
The second formula in the second picture should not have a n on the left. The third and fourth formula requires that $zj$ are not semi-integer rather than integer.