We start with a definition of a series of functions and look at its convergence and uniform convergence. Then we address the problem of preservation of properties.

The idea of function series comes naturally when we work with Taylor
polynomials. For reasonable functions, increasing the degree of
*T*_{n} improves quality of approximation, so it seems
that if we could somehow create an infinite Taylor polynomial, we would get
precisely the original function. An infinite Taylor polynomial is an
infinite sum of powers - a series. In this and subsequent sections we will
show that this idea of infinite Taylor polynomial actually does work - but
it takes some effort to make it work well.

When developing the notion of series for functions, we naturally take inspiration from series of numbers.

Definition.

By aseries of functionswe mean the symbolwhere

{ is some sequence of functions.f_{k}}_{k ≥ n0}Given a series of functions, we define its

region of convergenceas

{ xreal; ∑f_{k}(x) converges}.We define its

region of absolute convergenceas

{ xreal; ∑f_{k}(x) converges absolutely}.

Note that the definition makes sense. If we take some real number *x*
that is in domains of all *f*_{k} and substitute it into
all these functions, then
*f*_{k}(*x*)*x*, put it into the
function series and ask about the convergence of the resulting real series,
we usually cannot easily answer since this real series often changes signs.
Thus the natural thing to do is to inquire about absolute convergence, when
all the nice tests are available, moreover, absolute convergence allows us
to manipulate this series (see Theory - Introduction -
Absolute convergence).

Since for every *x* from the region of convergence we get a number - the
sum of the series *f*_{k}(*x*)*f* defined on this region of convergence. We call this
function the **sum** of the given series. However, in order to explore
this idea a bit further it is a good idea to change the point of view. We
used the "point by point" approach above, because it is more elementary and
above all because it allowed us to define those two regions. Now we use a
more abstract point of view, when we treat functions as objects, and create
the notion of series out of the notion of sequence of functions. This is
somewhat "cleaner" and more elegant, also it allows us to introduce uniform
convergence and convergence on a set. As usual, since convergence does not
depend on the beginning of the series, we will just write a sum, without
specifying limits. Of course, when we work with the sum of such a series,
then we need to know where the summation starts.

Definition.

Consider a series of functions∑ Assume thatf_{k}.fis a function defined on some setMand that allf_{k}are also defined on thisM.We say that this series

converges(pointwise)tofonMif the sequence ofpartial sumsconverges to

fonM. We writeWe say that this series

converges tofuniformly onMif the sequence{ of partial sums converges tos_{N}}funiformly onM. We write

Of course, usually we want uniform convergence and convergence on the largest set possible, in the latter case on the region of convergence of course. Since convergence of series is based on convergence of sequences (of partial sums) and there uniform convergence implies pointwise convergence, we immediately obtain the following.

Proposition.

If∑ on some setf_{k}⇉fM, then also∑ there.f_{k}=f

We should actually properly write limits with the sums in this Proposition,
since the concrete function *f* definitely depends on where
the indexing starts, that is, what functions are included in the sum. For
the sake of brevity we will usually skip it.

Before we show an example, we state a series version of a criterion that proved useful in case of sequences.

Proposition.

Consider a series of functions∑ This series converges to some functionf_{k}.funiformly on a setMif and only if

**Example:** Investigate the series
.

Its terms *x*^{k} are defined for all real numbers, so
that's where we start. Let *x* be some fixed real number. If you are
not used to playing like this with *x*, call this fixed number
*c*. Then we consider the series
*c*^{k}.*c*| < 1.

Next we note that if *x* is from *x*|*x*|^{k} = ∑ |*x*^{k}|*x*. Thus we know that

Now we would like to know whether and where this convergence is uniform.
Recall that for a series with sum *f*, "uniform convergence on a
set *M*" means the following: Given a tolerance, we should be able to
find a partial sum of the series that approximates *f* on *M* well
within that tolerance. To get some hint we first look at a picture
of *f* and several partial sums of the given series.

It seems that we may have trouble near 1. Indeed, we can say even without
any calculations that we definitely do not have uniform convergence on the
whole set *x* goes to 1 from the left, the
function *f* (*x*) = 1/(1-*x*)

The fact that the supremum is infinity can be easily seen by taking limit of
the expression inside at 1 from the left. Since all these suprema are
infinity, there is no way they can converge to 0 when we send *N* to
infinity.

We see that the real trouble is at 1 and also at −1 things do not look too
good (we already discussed this in the previous section, uniform convergence
is shaky at ends of the set of convergence), so what happens if we cut these
ends away? Consider some small positive number *d*, let
*M* = [−1 + *d*,1 − *d*].*M*. We check on the
supremum:

We again used the knowledge of geometric series and the fact that
*d*| < 1.

This example is very instructive. In order to study uniform convergence we
need to know *f*, but the global approach does not help in finding it.
Thus one first needs to use the pointwise approach: Fix *x* and
investigate absolute convergence of the resulting real series, usually using
the Root test or the Ratio test. If we are lucky, we find that this real
series converges to some number
*A* = *f* (*x*).*f* and then we can inquire about uniform
convergence. After all, this is exactly how it was with sequences of functions.

While in theory the situation for sequences and series is similar, in fact
here we have a serious problem. We know that deciding convergence of a
series is often relatively easy, so we can expect to find region of
convergence fairly often. However, determining the sum of a series is in
most cases a very difficult task, so finding *f* cannot be expected. How
do we then test uniform convergence? The most popular answer is the
following statement.

Theorem(the Weierstrass theorem).

Consider a series of functions∑ and a setf_{k}M. Assume that there are real numbersa_{k}such that∑ converges anda_{k}

| for allf_{k}(x)| ≤a_{k}kand allxfromM.Then the series

∑ converges uniformly onf_{k}M.

Using this theorem we easily show uniform convergence in the above example.
Indeed, for *x* from
*d*,1 − *d*]*x*^{k}| ≤ (1 − *d*)^{k}*d*)^{k}

**Example:** Investigate the series
.

When we fix some real *x*, then it becomes a constant and we can
compare the resulting series
of real numbers to a known
*p*-series to prove
that it converges.

However, we have no idea what the sum of such a series with a fixed *x*
is. Thus we know that the region of convergence of the given series of
functions is the whole real line, but we do not know its sum *f* and the
approach to uniform convergence via supremum is impossible.

However, the Weierstrass theorem works just fine, we just take *M* to
be the set of all real numbers and the above estimate suggests that
*a*_{k} = 1/*k*^{2}*k* and every real number *x* we have
*k*^{2} + *x*^{2}) ≤ *a*_{k}.

We still do not know to what function it converges, which may seem funny, but in fact just knowing that uniform convergence is taking place allows us to manipulate this series in many useful ways, as we will see below.

Theorem(the Dirichlet uniform convergence test).

Consider a series of functions∑ whose partial sums are uniformly bounded on a setf_{k}M. That is, there exist a real numberhsuch that for allxfromMand for allNwe haveLet

{ be a sequence of functions that converges uniformly to 0 ong_{k}}M. Then the series∑ converges uniformly onf_{k}g_{k}M.

**Example:** Investigate the series
.

When we fix some
*x* ≥ 0,*M* = [0,∞)*M* *is* the region of convergence,
because this depends on where the given series starts its indexing. For
instance, if indexing starts at
*n*_{0} = 3,*x* > −3*f*_{3} and thus it is out. We see that some
negative integers are ruled out and unless we know where the indexing
starts, we do not know which integers are out. Thus it is better to play it
safe and stick with non-negative numbers.

Now we would like to show uniform convergence on *M*, but the
Weierstrass theorem is no good here. We have a natural estimate
*k* + *x*) ≤ 1/*k*,*k*

We recall that the alternating
series with terms ^{k}*f*_{k}(*x*) = (−1)^{k}*M*, they become a series of functions whose
partial sums are uniformly bounded. The above estimate also shows that the
sequence of functions
*g*_{k}(*x*) = 1/(*k* + *x*)*M*. The assumptions of the Dirichlet test
are satisfied and we can conclude that the given series converges uniformly
on *M*.

Theorem(the Abel uniform convergence test).

Consider a decreasing sequence of functions{ that are uniformly bounded on some bounded closed intervalf_{k}(x)}M; that is, there existshsuch that| for allf_{k}(x)| ≤hkand allxfromM.

Let∑ be a convergent series of real numbers. Then the seriesa_{k}∑ converges uniformly ona_{k}f_{k}M.

We conclude this section with a consequence of Dini's theorem from the previous section. Note that if all functions in a given series are non-negative, then its partial sums form a non-decreasing (and therefore monotone) sequence of functions.

Theorem(Dini's theorem).

Consider a series of functions∑ that converges to somef_{k}fon some bounded closed intervalM. If all functionsf_{k}are non-negative onM, then the convergence of this series is uniform there.

We defined convergence of series using convergence of sequences, so everything that was stated about properties in the previous section carries over. As can be expected, series behave well when we apply the usual algebraic operations.

Theorem.

Assume that a series of functions∑ converges to a functionf_{k}fon a setMand that a series of functions∑ converges tog_{k}gon the same setM. Then the following are true:

(i) For any real numbera, the series∑ ( converges toa⋅f_{k})on a⋅fM.

(ii) The series∑ ( converges tof_{k}+g_{k})on f+gM.

(iii) The series∑ ( converges tof_{k}−g_{k})on f−gM.

(iv) The series∑ ( converges onf_{k}⋅g_{k})M.

Just like with real series, the pointwise multiplication in (iv) is not very useful and we prefer to multiply function series in a different way (Cauchy multiplication). We will return to this in section on power series.

The properties that were preserved for sequences are naturally also preserved here.

Theorem.

Assume that a series of functions∑ converges to a functionf_{k}fon a setM.

(i) If allf_{k}are odd, then alsofis odd.

(ii) If allf_{k}are even, then alsofis even.

(iii) If allf_{k}areT-periodic, then alsofisT-periodic.

(iv) If allf_{k}are non-decreasing functions, then alsofis a non-decreasing function.

(v) If allf_{k}are non-increasing functions, then alsofis a non-increasing function.

(vi) If allf_{k}are constant functions, then alsofis a constant function.

We know from previous section that we cannot expect preservation of continuity and other "better" properties from mere pointwise convergence (and it is easy to make up examples about it similar to those in the previous section), which is why we need uniform convergence.

Theorem.

Consider a series of functions∑ converging to a functionf_{k}f.

(i) If allf_{k}are continuous on a setMand∑ converges uniformly tof_{k}fonM, thenfis also continuous onM.

(ii) If allf_{k}are continuous on a setMand∑ converges uniformly tof_{k}fonM, then for every interval[ that is a subset ofa,b]Mone has(iii) Assume that all

f_{k}are continuous on an intervalMand that∑ converges uniformly tof_{k}fonM. Fix someafromMand forxfromMdefineThen

∑ converges uniformly toF_{k}FonM.(iv) If all

f_{k}are continuously differentiable on a setMand the series of their derivatives∑ converges uniformly to some functionf_{k}′gonM, thenfis differentiable onMandf′ =g.

Moreover,∑ actually converges tof_{k}funiformly onM.

Again we see that derivatives can be quite
tricky, even uniform convergence of the series of
*f*_{k} is not enough to get something reasonable and
one has to ask things about derivatives. As we discussed in the previous
section, these properties are in fact rules about exchange of order of
operations. We will show it in the next statement, which also shows that it
can be useful to know about uniform convergence without knowing its sum
and thus it justifies usefulness of the tests above. Since we are
just restating the above theorem, we decided to sacrifice precision of
statement to its clarity.

Proposition.

Assume that a series of functions∑ converges uniformly on an intervalf_{k}Mand also that∑ converges uniformly onf_{k}′Mif needed. Letaandbbelong toM. Then

Basic systems of functions

Back to Theory - Series of functions