Series o functions

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 a series of functions we mean the symbol

where { f_k}_k ≥ n0 is some sequence of functions.

Given a series of functions, we define its region of convergence as

{x real;  ∑ f_k(x)  converges}.

We define its region of absolute convergence as

{x real;  ∑ 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)  is a series of real numbers and therefore we can investigate its convergence and absolute convergence. Since we know that absolute convergence implies convergence, the region of absolute convergence is obviously a subset of the region of convergence. We will see later that in reasonable cases we get absolute convergence inside the region of convergence, only at its border things get worse. This is not all that strange, since the boundary of region of convergence is exactly the place where something goes wrong with convergence, so it would be too much to expect absolute convergence there. On the other hand, once we move inside the region of convergence, we are far from places where things go wrong and we usually get a "better" convergence out of it. Note that from a practical point of view the absolute convergence is fundamentally better. When we fix some real 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) - we obtain a certain function 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  ∑ f_k. Assume that f is a function defined on some set M and that all f_k are also defined on this M.

We say that this series converges (pointwise) to f on M if the sequence of partial sums

converges to f on M. We write

We say that this series converges to f uniformly on M if the sequence {s_N} of partial sums converges to f uniformly on M. 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  ∑ f_k ⇉ f on some set M, then also  ∑ f_k = f there.

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  ∑ f_k. This series converges to some function f uniformly on a set M if 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. This is a geometric series, so we know that it converges exactly if |c| < 1. That is, we know that the given series converges only when we take real numbers from the interval (−1,1). Using proper language, the set (−1,1) is the region of convergence of the given series.

Next we note that if x is from (−1,1), then also |x| is from this set, therefore we get convergence for the series  ∑ |x|^k = ∑ |x^k| and we have absolute convergence at this x. Thus we know that (−1,1) is also the region of absolute convergence. Moreover, here in this example we actually know what the sum is:

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 (−1,1). When x goes to 1 from the left, the function  f (x) = 1/(1-x) runs away to infinity, so it cannot be approximated uniformly by our partial sums - these are polynomials and therefore bounded on bounded sets. In order to confirm this, we look at appropriate suprema. We use the well-known formula for partial sums of geometric series.

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]. We claim that we have uniform convergence on M. We check on the supremum:

We again used the knowledge of geometric series and the fact that |1 − 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). In this way we obtain 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  ∑ f_k  and a set M. Assume that there are real numbers a_k such that  ∑ a_k  converges and

| f_k(x)| ≤ a_k for all k and all x from M.

Then the series  ∑ f_k  converges uniformly on M.

Using this theorem we easily show uniform convergence in the above example. Indeed, for x from [−1 + d,1 − d] we have |x^k| ≤ (1 − d)^k and numbers (1 − d)^k form a convergent geometric series.

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/k² will do the trick - and it indeed does. These numbers form a convergent series and for every k and every real number x we have 1/(k² + x²) ≤ a_k. By the Weierstrass theorem, the given series converges uniformly on the real line.

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.

Example: Investigate the series  .

When we fix some x ≥ 0, then it becomes a constant and we can use the Alternating series test to show that the resulting series converges. Thus the set M = [0,∞) is definitely a subset of the region of convergence, but as usual we have no idea what the sum of the given series is. By the way, not that we do not say that M is the region of convergence, because this depends on where the given series starts its indexing. For instance, if indexing starts at n₀ = 3, then also x > −3 are fine, but −3 is not in the domain of f₃ 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 1/(k + x) ≤ 1/k, but terms 1/k do not form a convergent series and obviously it is not possible to improve this estimate. Thus we turn to the Dirichlet test.

We recall that the alternating series with terms (−1)^k has bounded partial sums (partial sums are either 0 or 1 or −1, depending on where we start indexing), so when we take f_k(x) = (−1)^k as constant functions on 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) converges uniformly to 0 on M. The assumptions of the Dirichlet test are satisfied and we can conclude that the given series converges uniformly on 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  ∑ f_k  that converges to some f on some bounded closed interval M. If all functions f_k are non-negative on M, 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  ∑ f_k  converges to a function f on a set M and that a series of functions  ∑ g_k  converges to g on the same set M. Then the following are true:
(i)  For any real number a, the series  ∑ (a⋅ f_k)  converges to a⋅ f on M.
(ii)  The series  ∑ ( f_k + g_k)  converges to f + g on M.
(iii)  The series  ∑ ( f_k − g_k)  converges to f − g on M.
(iv)  The series  ∑ ( f_k⋅g_k)  converges on 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  ∑ f_k  converges to a function f on a set M.
(i)  If all f_k are odd, then also f is odd.
(ii)  If all f_k are even, then also f is even.
(iii)  If all f_k are T-periodic, then also f is T-periodic.
(iv)  If all f_k are non-decreasing functions, then also f is a non-decreasing function.
(v)  If all f_k are non-increasing functions, then also f is a non-increasing function.
(vi)  If all f_k are constant functions, then also f is 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.

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  ∑ f_k  converges uniformly on an interval M and also that  ∑ f_k′  converges uniformly on M if needed. Let a and b belong to M. Then

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