Let $f(z) = \sum\limits_{n=0}^\infty a_n z^n$ and let $g(z) = \sum\limits_{n=0}^\infty b_n z^n$. I need to show that $\sum\limits_{n=0}^\infty a_n z^n \sum\limits_{n=0}^\infty b_n z^n$ converges to $\sum\limits_{n=0}^\infty c_n z^n$ where each $c_n$ is given by $c_n = \sum\limits_{k=0}^n a_k b_{n-k}$.
This is not homework by the way.
The question says to use the Cauchy integral theorem and look at $f(z)g(z)/z^n = \sum\limits_{k=0}^\infty a_k z^{k-n} g(z)$, but I don't understand why it tells me to proceed this way.