Infinite sequences of functions were discussed earlier. Remember, there were two kinds of convergence, pointwise and uniform. As was just done for series of numbers, once you understand sequences, it is no problem to consider series. In this case, series of functions.
Definition 10.7.1 Let
whenever the limit exists. Thus there is a new function denoted by
and its value at x is given by the limit of the sequence of partial sums in 10.8. If for all x ∈ D, the limit in 10.8 exists, then 10.9 is said to converge pointwise. ∑ k=1∞fk is said to converge uniformly on D if the sequence of partial sums,
Proof: The first part follows from Theorem 10.1.8. The second part follows from observing the condition is equivalent to the sequence of partial sums forming a uniformly Cauchy sequence and then by Corollary 4.9.5, these partial sums converge uniformly to a function which is the definition of ∑ k=1∞fk. ■
Is there an easy way to recognize when 10.10 happens? Yes, there is. It is called the Weierstrass M test.
Proof: Let z ∈ D. Then letting m < n
whenever m is large enough because of the assumption that ∑ n=1∞Mn converges. Therefore, the sequence of partial sums is uniformly Cauchy on D and therefore, converges uniformly to ∑ k=1∞fk on D. ■
Proof: This follows from Theorem 4.9.3 applied to the sequence of partial sums of the above series which is assumed to converge uniformly to the function ∑ k=1∞fk. ■
There is no good reason not to present the following wonderful result of Weierstrass. It turns out, that every continuous function can be uniformly approximated by a polynomial.