It is shown that a power series that converges on an open interval defines an infinitely differentiable function on that interval. We give sufficient conditions for the limit of a sequence of functions or the sum of an infinite series of functions to be continuous, integrable, or differentiable. Cauchy’s uniform convergence criteria for sequences and series are proved, as is Dirichlet’s test for uniform convergence of a series. SECTION 4.4 deals with pointwise and uniform convergence of sequences and series of functions. For general series, we consider absolute and conditional convergence, Dirichlet’s test, rearrangement of terms, and multiplication of one infinite series by another. In connection with series of positive terms, we consider the comparison test, the integral test, the ratio test, and Raabe’s test. We prove Cauchy’s convergence criterion for a series of constants. SECTION 4.3 introduces concepts of convergence and divergence to \(\pm\infty\) for infinite series of constants.Continuity and boundedness of a function are discussed in terms of the values of the function at sequences of points in its domain. Limit points and boundedness of a set of real numbers are discussed in terms of sequences of members of the set. We show that if a sequence converges to a limit or diverges to \(\pm\infty\), then so do all subsequences of the sequence. SECTION 4.2 defines a subsequence of an infinite sequence.We prove the Cauchy convergence criterion for sequences of real numbers. The limit inferior and limit superior of a sequence are defined. We discuss bounded sequences and monotonic sequences. The concept of a limit of a sequence is defined, as is the concept of divergence of a sequence to \(\pm\infty\). SECTION 4.1 introduces infinite sequences of real numbers.IN THIS CHAPTER we consider infinite sequences and series of constants and functions of a real variable.
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