FOURTEEN - SEQUENCES AND SERIES

Abstract

This chapter discusses what is meant by an infinite sum. The infinite sums are called infinite series of real. The chapter discusses the theory of infinite series and how it can give a great deal of information about a wide variety of functions. A sequence of real numbers is a function of which domain is the set of positive integers. The values taken by the function are called terms of the sequence. The chapter presents the calculation of the limit of a sequence. There are certain kinds of sequences that have special properties worthy of mention. These sequences are bounded and monotonic sequences. A bounded monotonic sequence is convergent. Every unbounded sequence is divergent. The chapter discusses geometric series. The sum of a geometric progression is the sum of a finite number of terms. The chapter discusses how the geometric series can be used to resolve the paradox of Zeno. In general, one can use the geometric series to write any repeating decimal in the form of a fraction by using some technique. The rational numbers are exactly those real numbers that can be written as repeating decimals. The chapter discusses a great number of infinite series with examples such as harmonic series, infinite series of nonnegative terms, and alternating series. It also describes series of functions, such as power series, Taylor series, and Maclaurin series. The chapter discusses the differentiation and integration of power series.