Chap11_Sec1 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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Unformatted text preview: 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zenos paradoxes and the decimal representation of numbers. INFINITE SEQUENCES AND SERIES Their importance in calculus stems from Newtons 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. INFINITE SEQUENCES AND SERIES We will pursue his idea in Section 11.10 in order to integrate such functions as e- x 2 . Recall that we have previously been unable to do this. INFINITE SEQUENCES AND SERIES Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series. It is important to be familiar with the basic concepts of convergence of infinite sequences and series. INFINITE SEQUENCES AND SERIES Physicists also use series in another way, as we will see in Section 11.11 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. INFINITE SEQUENCES AND SERIES 11.1 Sequences In this section, we will learn about: Various concepts related to sequences. INFINITE SEQUENCES AND SERIES SEQUENCE A sequence can be thought of as a list of numbers written in a definite order: a 1 , a 2 , a 3 , a 4 , , a n , The number a 1 is called the first term , a 2 is the second term , and in general a n is the n th term . SEQUENCES We will deal exclusively with infinite sequences. So, each term a n will have a successor a n +1 . SEQUENCES Notice that, for every positive integer n , there is a corresponding number a n . So, a sequence can be defined as: A function whose domain is the set of positive integers SEQUENCES However, we usually write a n instead of the function notation f ( n ) for the value of the function at the number n . SEQUENCES The sequence { a 1 , a 2 , a 3 , . . .} is also denoted by: { } { } 1 or n n n a a = Notation SEQUENCES Some sequences can be defined by giving a formula for the n th term. Example 1 SEQUENCES In the following examples, we give three descriptions of the sequence: 1. Using the preceding notation 2. Using the defining formula 3. Writing out the terms of the sequence Example 1 SEQUENCES In this and the subsequent examples, notice that n doesnt have to start at 1. Example 1 a Preceding Notation Defining Formula Terms of Sequence 1 1 n n n = + 1 n n a n = + 1 2 3 4 , , , ,..., ,......
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Chap11_Sec1 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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