Sequences and Series of Functions

This chapter extends the work on sequences and series to a new dimension by considering sequences and series of functions. The first type of convergence that one wants to consider is called pointwise convergence. This definition asks us to consider the convergence of a sequence of functions on a shared domain by examining each point separately and seeing whether the resulting sequence of numbers converges. Pointwise convergence preserves some properties of functions (such as nonnegativity), while it fails to preserve other properties (such as convergence). A pointwise limit of a sequence might not inherit some of the properties that the sequence functions had. A uniformly convergent sequence of functions will automatically be pointwise convergent. Uniform convergence is a stronger condition than pointwise convergence, in the sense that every uniformly convergent sequence of functions is also pointwise convergent, but the converse is not true.