ABSTRACT
This chapter describes a type of sequence of functions: the partial sums that is got by adding up the functions from a sequence. One cannot argue directly for the continuity of the pointwise limit function; one must instead approach the issue indirectly by looking at convergence on compact subsets of the domain of the function which can grow to include any arbitrary point in the domain of the function. The chapter proves a variant of the Weierstrass test for uniform convergence and looks at the problem of differentiating a series of functions. The series obtained by differentiating the derived series term by term is called the second derived series. The third derived series is the series derived from the second derived series and so forth.
