Let \((a_n)\) be a sequence in \(\mathbb{R}\) (or \(\mathbb{C}\)) and \(z_0\in \mathbb{R}\) (or \(\mathbb{C}\)). The function series \(\sum_{n=0}^{\infty} a_n(z-z_0)^n\) is called power series. The point \(z_0\) is called center or development point and \((a_n)\) are the coefficients of the power series.
Power series can be also defined on \(\mathbb{R}^n\) or \(\mathbb{C}^n\). Using multi-index notation we write a power series as
\begin{equation*} \sum_{\mu\in \mathbb{N}^n} a_\mu (z-z_0)^\mu. \end{equation*}Operations creating new power series Link to heading
Remarks
- For every one-dimensional power series there is a radius of convergence .
- Every power series is continuous.
- Every power series is smooth.
- Every power series is analytic.