Find the Laurent expansion of $1/(\cos(z)-1)$ in the regions $|z|<2\pi$ and $2\pi<|z| <4\pi$

Question

Find the terms of Laurent expansion of $\frac{1}{\cos(z)-1}$ valid for the regions:
a) $|z|<2\pi$,
b) $2\pi<|z| <4\pi$.

I tried to find the coefficients of Laurent series using the contour integral formula for $B_n$'s but its all getting messy. Please suggest a way out.

Answer 1

Since $$\cos(x)= 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}.....$$ Your series is $$\frac{1}{-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}.....}$$ Now you can do long division and find the coefficient of the laurent series. long division is allowed since product (or division) of two convergent series is convergent.