proving ring of convergence $fg=\sum\limits_{m,n=-\infty}^{\infty} a_n b_m (z-z_0)^{n+m}$

Question

how to prove this

I am stuck at this point $$fg=\sum_{m,n=-\infty}^{\infty} a_n b_m (z-z_0)^{n+m}=\sum_{i=-\infty}^{\infty} c_i(z-z_0)^{2n}$$ what to do next ? anyone please help me in proving this

Answer 1

It would help to write the series as $\sum\limits_{n=-\infty}^\infty c_n (z-z_0)^n$ and find $c_n$ (hint: they're given by infinite sums; it might help to first consider the finite case, then generalize). For the convergence, you might find that the following result helps:

Proposition: If $A=\sum\limits_{n=-\infty}^\infty \alpha_n$ and $B=\sum\limits_{n=-\infty}^\infty \beta_n$ both converge absolutely, then so does their product $AB.$