$$\int_{|z|=1}\frac{cosz}{z^5}dz$$
$$\frac{cosz}{z^5}=\frac{\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{2n!}}{z^5}=\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{2n!\cdot z^5}=\frac{1}{z^5}-\frac{1}{2z^3}+\frac{1}{4!z}-\frac{z}{6!}+\frac{z^3}{8!}$$
How do I find $c_{-1}$?
$$\int_{|z|=1}\frac{cosz}{z^5}dz = 2i\pi f^{(4)}(0)/4! =2i\pi/4! $$ where $f(z) = \cos (z)$ See here Calculation of $\oint_{\vert z\vert=1}\frac{e^{z^2+\sin(z)}}{4(z-2)^2e^{\cos(z)}}dz$.