Why is the composition of convergent Laurent series convergent?

Question

In Rick Miranda's book Algebraic Curves and Riemann Surfaces, the author proves the order of a meromorphic function $f \colon X \to \mathbf{C}$ at a point $p$ is independent of the complex chart that is used. To do so, he chooses charts $\phi \colon U \to V$ and $\psi \colon U' \to V'$ and substitutes the Taylor series expansion of the transition function $\phi \circ \psi^{-1}$ at $\psi(p)$ into the Laurent series expansion of $f \circ \phi^{-1}$ at $\phi(p)$. I was wondering why such a substitution produces a convergent series.

Answer 1

The Laurent series of a meromorphic function is just a power series with a finite number of negative exponents. That is, a power series multiplied by negative power of the variable. This does not affect the convergence of the series.