Optimization and Control Theory Department: 2011-06-14 Seminarium

Xiao-Xiong Gan
(Mathematics Department, Morgan State University, Baltimore, USA)

Formal Power Series and the Boundary Convergence of Power Series

Uwaga: seminarium odbędzie się w sali A1-33 (sala posiedzeń Rady Wydziału o godzinie 9.00.

The behaviour of power series on boundary of convergence domain has been an interesting topic since power series was introduced. For example,

$f(x) = \sum_{n=1}^{\infty}\dfrac{(-1)^n}{n}x^n$

converges on $(-1,1]$ but diverges at $x = -1$ , and $f^{\prime}(x)$ diverges at both $x = 1$ and $x = -1$ . The composition of formal power series has been an important part of the formal power series theory. We introduce some relationship between these two subjects and provide a condition for convergence of a power series at every point in its interval of convergence, including endpoints or boundary points.