where denotes the partial sum of the series.

A famous example of a non-uniformly convergent sequence of functions is given in Calculus: take oj the interval . The limit of this sequence is the function such that for and . As this limit is non continuous at 1, the sequence is not uniformly convergent (see Fig. 1).

For the series , it is defined on , but convergent only on .

As a consequence, we have that power series converge uniformly on their open domain of convergence (v.i. 0.8).

Noah Dana-Picard 2007-12-24