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 10.7.1Let

{fn}

be a sequence of functions defined on D.Then

( ∞ ) n
∑ fk (x) ≡ lim ∑ fk(x) (10.8)
k=1 n→∞ k=1

(10.8)

whenever the limit exists. Thus there is a new function denoted by

∞∑
fk (10.9)
k=1

(10.9)

and its value at x is given by the limit of the sequence of partial sums in 10.8. If for allx ∈ D, the limit in 10.8exists, then 10.9is said to converge pointwise.∑_{k=1}^{∞}f_{k}is saidto converge uniformly on D if the sequence of partial sums,

{∑nk=1fk}

_{n=1}^{∞}convergesuniformly.If the indices for the functions start at some other value than 1, you make theobvious modification to the above definition as was done earlier with series ofnumbers.

Theorem 10.7.2Let

{fn}

be a sequence of functions defined on D.The series∑_{k=1}^{∞}f_{k}converges pointwise if and only if for each ε > 0 andx ∈ D, there exists N_{ε,x}which may depend on x as well as ε such that whenq > p ≥ N_{ε,x},

| |
||∑q ||
|| fk(x)|| < ε
|k=p |

The series∑_{k=1}^{∞}f_{k}convergesuniformly on D if for every ε > 0 there exists N_{ε}suchthat if q > p ≥ N_{ε}then

| |
||∑q ||
sup || fk (x)||< ε (10.10)
x∈D |k=p |

(10.10)

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}^{∞}f_{k}. ■

Is there an easy way to recognize when 10.10 happens? Yes, there is. It is called the
Weierstrass M test.

Theorem 10.7.3Let

{fn}

be a sequence of functions defined on D.Suppose there exists M_{n}such that sup

{|fn (x)| : x ∈ D}

< M_{n}and∑_{n=1}^{∞}M_{n}converges. Then∑_{n=1}^{∞}f_{n}converges uniformly on D.

whenever m is large enough because of the assumption that ∑_{n=1}^{∞}M_{n} converges.
Therefore, the sequence of partial sums is uniformly Cauchy on D and therefore,
converges uniformly to ∑_{k=1}^{∞}f_{k} on D. ■

Theorem 10.7.4If

{fn}

is a sequence of functions defined on D which arecontinuous at z and∑_{k=1}^{∞}f_{k}converges uniformly, then the function∑_{k=1}^{∞}f_{k}must also be continuous at z.

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}^{∞}f_{k}.■

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.