Chapter 2. Functional series

2.1 The domain of convergence

Let a sequence of functions be given which are defined on

The same set . Series of the form

Are called functional, since it terms are functions of x.

Example 2.1. The terms of the series

Are defined on the interval and the terms of the series

Are defined on the interval

With fixed , we can obtain various number series, which may be convergent or divergent. The set of those values for which a functional series converges is called the domain of convergence of functional series.

Example 2.2. Find the domain of convergence of the series

Solution

The number series converges for and diverges for and so of we put , we obtain the given series, which will converge for , i.e. for , and diverge for , i.e. for . The interval of convergence is

A functional series is a certain function of in the domain of its convergence.

2.2. Types of convergence

We now consider a series of the form: whose terms are the functions defined over the same finite interval .

We say that series converges to a function on this interval if the deviation of the partial sum from tends to zero as if increases.

Depending on the form of the deviation we speak about the type of the convergence of series. For instance if

,

We say that the series uniformly converges to its sum .

Similarly, we say that the series converges to in the mean or in the mean square depending on whether we have

Or

.

Example 2.3. Show that the functional series converges uniformly on the interval

Solution

The series is alternating; it meets the condition of the Leibniz test, and hence it converges for any . Let be its sum, and be its th partial sum. The remainder

does not exceed the fist term in the absolute value , i.e. 1, for all and for all . Take any . The inequality will then hold if , hence .

2.3. The simplest properties of functional series

Property 1. If a series converges uniformly it also converges in the mean (of any order) to the same sum.

The converse statement may not be true in the general case.

The prove the Property we replace the difference by maximal deviation . The integrals can only increase and hence we obtain:

.

The inequality prove the statement.

Property 2. If the series uniformly converges on an interval we have for each number belonging to the interval.

This property makes it possible to obtain the sum of numerical series from a functional sum when its sum is known.

Example 2.4. The series Converges to for any , so that substitution we obtain:

Property 3. (Weierstrass’s test.)

To test a series for uniformly convergence we usually apply Weierstrass’s test whose condition is sufficient for the uniformly convergence: if all satisfy the inequalities and the number series converges, then the series converges uniformly as well.

The proof. Let us consider the deviation:

.

Since the last sum is the reminder of a converging series it tends to zero as

The number series is often called the dominant series for the functional series.

Example 2.5. The series converges uniformly, because we have the inequalities and the series

is convergent .

Remark. The functional series can also converge uniformly on when there is no dominant number series, i.e. the Weierstrass’s test is only a sufficient test for uniform convergence, but not necessary one.

Example 2.6. It has been shown above (Example 2.3.), converges uniformly on the interval ,but for it there is no dominant convergent number series. In fact, the inequality holds for all natural and all and the number series diverges.