This process is experimental and the keywords may be updated as the learning algorithm improves. These keywords were added by machine and not by the authors. At the end of Section 9.2, we state and prove an important result due to Weierstrass, which in a simple form states that “any continuous function on can be uniformly approximated by polynomials.” Keywords Finally, we also discuss the Abel summability of series. At the end of the section, we also include some foundations for the study of summability of series, which is an attempt to attach a value to a series that may not converge, thereby generalizing the concept of the sum of a convergent series. In Section 9.2, we discuss a characterization for interchanging limit and integration signs, and interchange of limit and differentiation signs for uniform convergence of sequences and series of functions. Together they say that if, in addition to the above conditions, the set A is a topological space and the functions f n are continuous on A, then the series converges to a continuous function. In addition, we present characterizations for interchanging limit and integration signs in sequences of functions. The result is often used in combination with the uniform limit theorem. Our particular emphasis in Section 9.1 is to present the definitions and simple examples of pointwise and uniform convergence of sequences. Meta ( Main Idea) Pointwise convergence is again a direct application of our basic knowledge regarding sequences of real numbers. Watson, "A course of modern analysis", Cambridge Univ.In this chapter we consider sequences and series of real-valued functions and develop uniform convergence tests, which provide ways of determining quickly whether certain sequences and infinite series have limit functions. The functions f n ( x) x n ( 1 x) converge both pointwise and uniformly to the zero function as n. ![]() Rudin, "Principles of mathematical analysis", McGraw-Hill (1976) Nikol'skii, "A course of mathematical analysis", 1–2, MIR (1977) (Translated from Russian) Kudryavtsev, "A course in mathematical analysis", 1–2, Moscow (1981) (In Russian) Poznyak, "Fundamentals of mathematical analysis", 1–2, MIR (1971–1973) (Translated from Russian) Dieudonné, "Foundations of modern analysis", Acad. Cauchy, "Analyse algébrique", Gauthier-Villars (1821) (German translation: Springer, 1885) Corresponding statements can be easily generalized to series of (bounded or continuous) functionsĪ generalization of the concept of Cauchy sequence is that of Cauchy filter in a Uniform space, to which a corresponding notion of completeness is attached.Ī.L. A widely used special case of this theorem is when $X$ is a subset of $\mathbb R^n$ or, in particular, an interval $I\subset \mathbb R$ (cf. Therefore we also conclude that a Cauchy sequence of bounded continuous functions converges uniformly to a bounded continuous function. More precisely it states thatĪ sequence $\ (X), d)$. ![]() ![]() From this, and 3) (note that 3) is saying that the sequence (fn) converges uniformly to 0 ), it will follow that the series is uniformly Cauchy on D and thus uniformly convergent on D. The Cauchy criterion is a characterization of convergent sequences of real numbers. Essentially, such a series is alternating: for each x D, the series n 1( 1)n + 1fn(x) is a convergent alternating series. The elementary Cauchy criterion for sequences of real numbers The text also assumed the converse, which is that there exist a constant and a sequence and subsequence such that. In summary, the text proved that if converges uniformly to some then is uniformly Cauchy. ![]() 4.1 Cauchy criterion for uniform convergence Cauchy Convergence Functions Sequence Uniform Uniform convergence.4 Complete metric spaces and Banach spaces.2.3 Cauchy criterion for improper integrals.2.2 Cauchy criterion for real-valued functions.1 The elementary Cauchy criterion for sequences of real numbers.
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