For every power series
\(P(z)=\sum_{n=0}^{\infty} a_n(z-z_0)^n\) there is a unique number \(r\in [0,\infty]\), such that \(P\) converges locally absolutely uniformly
on \(D_r(z_0)=\lvert z-z_0\rvert
We call \(D_r(z_0)\) domain of convergence.
If \(r=0\) we say \(P\) is nowhere convergent. For \(r=\infty\) it is everywhere convergent.
Remarks
- One cannot say anything about the boundary of \(D_r(z_0)\) in general.
- The radius of convergence can be calculated using Cauchy-Hadamard theorem or the ratio test .
- In some sense \(\lvert a_n\rvert\) behaves like \(r^{-n}\), i.e. for \(0