Convergent region pf a Laurent series

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)

• \begin{align} \sum_{-\infty}^\infty a_{|k|}z^k &= \sum_1^\infty a_{|k|}z^k + \sum_{-\infty}^0 a_{|k|}z^k \\&= \sum_1^\infty a_kz^k + \sum_0^\infty a_k z^{-k} \\&= -a_0 + \sum_0^\infty a_kz^k + \sum_0^\infty a_k z^{-k} \\&= -a_0 + \sum_0^\infty a_k \big(z^k + z^{-k} \big) \\&= a_0 + \sum_1^\infty a_k \big(z^k + z^{-k} \big) \end{align} – Brevan Ellefsen Apr 23 at 5:24

The series is nothing but $$f(z)+ \sum\limits_{n=1}^{\infty} a_n\frac 1 {z^{n}}$$. It converges if $$|z| and $$|\frac 1 z| ie, if $$\frac 1 R <|z|. The sum is $$f(0)+g(z)+g(\frac 1 z)=f(z)+f(\frac 1 z)-f(0)$$ (where $$g(z)=f(z)-f(0)$$).