tailieunhanh - Ebook Essential calculus - Early transcendentals (2nd edition): Part 2
(BQ) Part 2 book "Essential calculus - Early transcendentals" has contents: Series, parametric equations and polar coordinates, vectors and the geometry of space, partial derivatives, multiple integrals, vector calculus. | 8 SERIES Infinite series are sums of infinitely many terms. (One of our aims in this chapter is to define exactly what is meant by an infinite sum.) Their importance in calculus stems from Newton’s idea of representing functions as sums of infinite series. For instance, in finding areas he often integrated a function by first expressing it as a series and then integrating each term of the series. We will pursue his idea in 2 Section in order to integrate such functions as eϪx . (Recall that we have previously been unable to do this.) Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series, so it is important to be familiar with the basic concepts of convergence of infinite sequences and series. Physicists also use series in another way, as we will see in Section . In studying fields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represents it. SEQUENCES A sequence can be thought of as a list of numbers written in a definite order: a1 , a2 , a3 , a4 , . . . , an , . . . The number a 1 is called the first term, a 2 is the second term, and in general a n is the nth term. We will deal exclusively with infinite sequences and so each term a n will have a successor a nϩ1 . Notice that for every positive integer n there is a corresponding number a n and so a sequence can be defined as a function whose domain is the set of positive integers. But we usually write a n instead of the function notation f ͑n͒ for the value of the function at the number n. NOTATION The sequence {a 1 , a 2 , a 3 , . . .} is also denoted by ͕a n ͖ or ϱ ͕a n ͖ n 1 EXAMPLE 1 Some sequences can be defined by giving a formula for the nth term. In the following examples we give three descriptions of the sequence: one by using the preceding notation, another by using the defining formula, and a third by writing out
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