Problem with Laurent series

Question

I am trying to find a Laurent series for $\cos(\frac{1}{z})z$.

I know that $$ \cos(1/z) = \frac{1}{2} e^{(-i/z)}z + \frac{1}{2} e^{(i/z)}z = z \sum_{n=0}^\infty \frac{(-1)^n (\frac{1}{n})^{(2n)}}{(2n)!}. $$

I just don't know what I should do next

Answer 1

I assume you know the power series for $\cos(z)$. Now just plug $1/z$ instead of $z$ for a Laurent series for $\cos (1/z)$ (around $0$). Then multiply by $z$. Done.

Looking closely at what you wrote you actually have it almost right, when you write $$z \sum_{n=0}^\infty \frac{(-1)^n (\frac{1}{n})^{(2n)}}{(2n)!}$$ It should be $$z \sum_{n=0}^\infty \frac{(-1)^n (\frac{1}{z})^{(2n)}}{(2n)!}$$ Now, rewrite as
$$ \sum_{n=0}^\infty \frac{(-1)^n (\frac{1}{z})^{(2n-1)}}{(2n)!}$$ and perhaps do some reindexing according to the conventions you are used to.