How do I find the region of convergence in the following case:
Let $f(z)=\sum_{i=0}^\infty a_iz^i$ be a power series with a positive radius of convergence. Determine the region of convergence of the Laurent series $\sum_{i=-\infty}^\infty a_{\vert i\vert}z^i$ and identify the function which it represents there.
The question in the book (Theory of complex functions, Remmert, page 360)
The series is nothing but $f(z)+ \sum\limits_{n=1}^{\infty} a_n\frac 1 {z^{n}}$. It converges if $|z| <R$ and $|\frac 1 z|<R$ ie, if $\frac 1 R <|z|<R$. The sum is $f(0)+g(z)+g(\frac 1 z)=f(z)+f(\frac 1 z)-f(0)$ (where $g(z)=f(z)-f(0)$).