Introduction to Analysis · Chapter 09

Sequences and series of functions

What survives a limit taken one point at a time, and what needs the stronger kind: continuity, integration, differentiation, the test that gives uniformity for free, and power series.

f_{n} ⇉ f

x \in E sup ∣ f_{n} (x) - f (x) ∣

n = 0 \sum \infty a_{n} x^{n}

∣ x ∣ < R

f_{n} (x) \to f (x)

\sum M_{n}

∥ f_{n} - f ∥_{\infty}

n = 1 \sum \infty n a_{n} x^{n - 1}

Lessons

Build the foundation

01Pointwise convergenceA sequence of functions can converge in the simplest way imaginable: fix a point, and the numbers at that point converge. Nothing else is asked, and nothing else is delivered. The limit function is perfectly well defined and almost every good property of the functions can fail to reach it — continuity, the value of the integral, the derivative. This lesson is the list of failures, and the next one is the condition that prevents them.Read lesson →
02Uniform convergenceMove the quantifier: demand a single stage beyond which every function of the sequence is within the tolerance at every point at once. That is uniform convergence, and it is exactly the repair the failures of the last lesson needed. The working test is a single number per stage — the largest distance between the function and the limit — and uniform convergence is that number tending to zero.Read lesson →
03Uniform limits of continuous functionsThe first of the three repairs. A uniform limit of continuous functions is continuous, and the proof is the argument every analyst knows by the number of times the tolerance is divided: a third for getting from the limit to a function of the sequence, a third for the continuity of that function, and a third for getting back. Uniformity is what makes the first and last thirds available at every point at once.Read lesson →
04Interchanging limits and integralsThe second repair. Under uniform convergence the integral of the limit is the limit of the integrals, and the proof is one line of estimate: the two integrals differ by at most the worst error times the length of the interval. Under pointwise convergence the statement is false, and the moving spike of the first lesson is the counterexample — its integral is one at every stage and zero in the limit.

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05Interchanging limits and derivativesThe third case, and the one that does not behave. Uniform convergence of the functions says nothing about their derivatives: a wiggle whose height shrinks may keep its steepness, and shrinking the height is all uniformity measures. The correct theorem therefore assumes uniform convergence of the derivatives, not of the functions, and asks only that the functions themselves converge at a single point.Read lesson →

06Series of functions and the M-testA series of functions is the sequence of its partial sums, so everything proved about sequences of functions applies to it at once. What is new is a way of establishing uniformity without ever computing a worst error: if each term is bounded by a number, and those numbers form a convergent series, then the series of functions converges uniformly. It is the comparison test of chapter five, made to deliver uniformity as well as convergence.Read lesson →

07Power series and the radius of convergenceA power series is a series of functions whose terms are the simplest possible: constants times powers. Their behaviour is remarkably tidy. There is a number, possibly zero and possibly infinite, such that the series converges absolutely inside that distance of the centre and diverges outside it, and on any closed interval strictly inside it converges uniformly. The number comes straight out of the root test of chapter five.Read lesson →

08Differentiating and integrating power seriesInside its range a power series behaves as though it were a polynomial: it may be differentiated and integrated term by term, and the resulting series has the same radius. That is a strong statement — differentiation was the operation uniform convergence failed to protect — and it is true here because the differentiated series is itself a power series whose radius can be computed and shown to be unchanged.Read lesson →

09Course reviewOne idea runs through the whole course: a tolerance is named and something must answer it. In chapter four the answer was a place along the index, in chapter six a radius about a point, in chapter eight a partition, and in chapter nine a stage of a sequence of functions. One axiom is behind all of them, and this lesson is where the shape of the whole thing is worth looking at.Read lesson →

Mapping the lesson.

Sequences and series of functions

Build the foundation