I am a student whose Complex Analysis' knowledge is gone for a long time. With this in mind, I start by apologizing if my question is too trivial.
Let us consider the complex function
$$ \frac{1}{1 + \sum_{l=1}^n c_l z^l},$$
where $c_l$ are some coefficients and $z$ is our variable. This function is clearly holomorphic in a neighbourhood of $z=0$ and thus it admits an expansion as a power series, i.e., there exists coefficients $a_n$ such that
$$ \frac{1}{1 + \sum_{l=1}^n c_l z^l} = \sum_{n=0}^\infty a_nz^n. $$
Now, my question is: In which domain can we guarantee convergence of this series? I know the follwing is a well-known result in complex analysis:
If a function $f$ is holomorphic at $D(0,R),$ then there exist coefficients $a_n$ such that $$ f(z) = \sum_{n=0}^\infty a_n z^n, \quad \text{ for }|z| < R. $$
So, to determine the domain of convergence of the series I presented above I believe that all one should do is to determine the largest disk centered in $0$ in which our function is holomorphic. I don't know how to do this thought.
Thanks for any help in advance.
Take $\displaystyle \gamma = \min \left\{ \frac{1}{n \max_{l \in \{1,\cdots,n\}} |c_l| },1\right\},$ so that $\displaystyle \left|\sum_l c_l z^l\right|<1$, for $|z|<\gamma$ and $f(z)$ is holomorphic and thus analytic there. That is $R$, the radius of convergence for a series centered at zero satisfies $R \ge \gamma.$