The Characteristic of Power Series and its Sum Function on the Convergence Circle

Abstract

In the paper, the specific issues is discussed whether or not the points on the convergence circle are the singular point of a sum function of a class of power series. Whats more, the relationship between divergence of the power series on the convergence circle and the pole of its function on the convergence circle is explored. And a new result is obtained that there exists the pole of its function on the convergence circle, the power series has the characteristic of everywhere divergence on the convergence circle.