(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 Section 8.7 in order to integrate such functions as eϪx (Recall that we have previously been unable to 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 8.8 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 8.1 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 is called the first term, a 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 , a , a , } is also denoted by ͕a n ͖ or ϱ ͕a n ͖ n1 EXAMPLE 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 the terms of the sequence Notice that n doesn’t have to start at (a) (b) (c) (d) 410 ͭ ͮ ͭ ͮ n nϩ1 ϱ an n nϩ1 an ͑Ϫ1͒n͑n ϩ 1͒ 3n n1 ͑Ϫ1͒n͑n ϩ 1͒ 3n {sn Ϫ } ϱn3 a n sn Ϫ , n ജ ͭ ͮ a n cos n cos ϱ n0 n , nജ0 ͭ ͭ ͮ n , , , , , , nϩ1 ͮ ͑Ϫ1͒n͑n ϩ 1͒ Ϫ , ,Ϫ , , , , 27 81 3n {0, 1, s2 , s3 , , sn Ϫ , } ͭ 1, n s3 , , 0, , cos , 2 ͮ ■ SECTION 8.1 V EXAMPLE SEQUENCES ■ 411 Find a formula for the general term a n of the sequence ͭ ͮ ,Ϫ , ,Ϫ , , 25 125 625 3125 assuming that the pattern of the first few terms continues SOLUTION We are given that a1 a2 Ϫ 25 a3 125 a4 Ϫ 625 a5 3125 Notice that the numerators of these fractions start with and increase by whenever we go to the next term The second term has numerator 4, the third term has numerator 5; in general, the nth term will have numerator n ϩ The denominators are the powers of 5, so a n has denominator n The signs of the terms are alternately positive and negative, so we need to multiply by a power of Ϫ1 In Example 1(b) the factor ͑Ϫ1͒ n meant we started with a negative term Here we want to start with a positive term and so we use ͑Ϫ1͒ nϪ1 or ͑Ϫ1͒ nϩ1 Therefore, a n ͑Ϫ1͒ nϪ1 nϩ2 5n ■ EXAMPLE Here are some sequences that don’t have a simple defining equation (a) The sequence ͕ pn ͖, where pn is the population of the world as of January in the year n (b) If we let a n be the digit in the nth decimal place of the number e, then ͕a n ͖ is a well-defined sequence whose first few terms are ͕7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, ͖ (c) The Fibonacci sequence ͕ fn ͖ is defined recursively by the conditions f1 f2 fn fnϪ1 ϩ fnϪ2 nജ3 Each term is the sum of the two preceding terms The first few terms are a¡ a™ a£ ͕1, 1, 2, 3, 5, 8, 13, 21, ͖ a¢ This sequence arose when the 13th-century Italian mathematician known as Fibonacci solved a problem concerning the breeding of rabbits (see Exercise 41) ■ A sequence such as the one in Example 1(a), a n n͑͞n ϩ 1͒, can be pictured either by plotting its terms on a number line as in Figure or by plotting its graph as in Figure Note that, since a sequence is a function whose domain is the set of positive integers, its graph consists of isolated points with coordinates FIGURE an ͑1, a1 ͒ ͑2, a2 ͒ ͑3, a3 ͒ ͑n, a n ͒ a¶= FIGURE n From Figure or it appears that the terms of the sequence a n n͑͞n ϩ 1͒ are approaching as n becomes large In fact, the difference 1Ϫ n nϩ1 nϩ1 412 ■ CHAPTER SERIES can be made as small as we like by taking n sufficiently large We indicate this by writing n lim 1 nlϱ n ϩ In general, the notation lim a n L nlϱ means that the terms of the sequence ͕a n ͖ approach L as n becomes large Notice that the following definition of the limit of a sequence is very similar to the definition of a limit of a function at infinity given in Section 1.6 DEFINITION A sequence ͕a n ͖ has the limit L and we write lim a n L a n l L as n l ϱ or nlϱ if we can make the terms a n as close to L as we like by taking n sufficiently large If lim n l ϱ a n exists, we say the sequence converges (or is convergent) Otherwise, we say the sequence diverges (or is divergent) Figure illustrates Definition by showing the graphs of two sequences that have the limit L an an L L FIGURE Graphs of two sequences with lim an= L 0 n n n ` A more precise version of Definition is as follows DEFINITION A sequence ͕an ͖ has the limit L and we write lim an L ■ Compare this definition with Definition 1.6.7 a n l L as n l ϱ or nlϱ if for every Ͼ there is a corresponding integer N such that if nϾN Խa then n Խ ϪL Ͻ Definition is illustrated by Figure 4, in which the terms a , a , a , are plotted on a number line No matter how small an interval ͑L Ϫ , L ϩ ͒ is chosen, there exists an N such that all terms of the sequence from a Nϩ1 onward must lie in that interval a¡ FIGURE a£ a™ aˆ aN+1 aN+2 L-∑ L L+∑ a˜ aß a∞ a¢ a¶ SECTION 8.1 SEQUENCES 413 ■ Another illustration of Definition is given in Figure The points on the graph of ͕an ͖ must lie between the horizontal lines y L ϩ and y L Ϫ if n Ͼ N This picture must be valid no matter how small is chosen, but usually a smaller requires a larger N y y=L+∑ L y=L-∑ FIGURE n N If you compare Definition with Definition 1.6.7, you will see that the only difference between lim n l ϱ a n L and lim x l ϱ f ͑x͒ L is that n is required to be an integer Thus we have the following theorem, which is illustrated by Figure THEOREM If lim x l ϱ f ͑x͒ L and f ͑n͒ a n when n is an integer, then lim n l ϱ a n L y y=ƒ L FIGURE x In particular, since we know that limx l ϱ ͑1͞x r ͒ when r Ͼ 0, we have lim nlϱ 0 nr if r Ͼ If a n becomes large as n becomes large, we use the notation lim n l ϱ a n ϱ The following precise definition is similar to Definition 1.6.8 DEFINITION lim n l ϱ a n ϱ means that for every positive number M there is an integer N such that if nϾN then an Ͼ M If lim n l ϱ a n ϱ, then the sequence ͕a n ͖ is divergent but in a special way We say that ͕a n ͖ diverges to ϱ The Limit Laws given in Section 1.4 also hold for the limits of sequences and their proofs are similar 414 ■ CHAPTER SERIES If ͕a n ͖ and ͕bn ͖ are convergent sequences and c is a constant, then Limit Laws for Sequences lim ͑a n ϩ bn ͒ lim a n ϩ lim bn nlϱ nlϱ nlϱ lim ͑a n Ϫ bn ͒ lim a n Ϫ lim bn nlϱ nlϱ nlϱ lim ca n c lim a n nlϱ lim c c nlϱ nlϱ lim ͑a n bn ͒ lim a n ؒ lim bn nlϱ nlϱ lim lim a n an nlϱ bn lim bn nlϱ nlϱ if lim bn nlϱ nlϱ [ lim a np lim a n nlϱ nlϱ ] p if p Ͼ and a n Ͼ The Squeeze Theorem can also be adapted for sequences as follows (see Figure 7) If a n ഛ bn ഛ cn for n ജ n and lim a n lim cn L, then lim bn L Squeeze Theorem for Sequences nlϱ cn nlϱ nlϱ Another useful fact about limits of sequences is given by the following theorem, whose proof is left as Exercise 45 bn an Խ Խ If lim a n 0, then lim a n THEOREM nlϱ nlϱ n FIGURE The sequence ͕ bn ͖ is squeezed between the sequences ͕ a n ͖ and ͕ cn ͖ EXAMPLE Find lim nlϱ n nϩ1 SOLUTION The method is similar to the one we used in Section 1.6: Divide numerator and denominator by the highest power of n that occurs in the denominator and then use the Limit Laws lim nlϱ n lim nlϱ nϩ1 This shows that the guess we made earlier from Figures and was correct ■ nlϱ ln n n 1ϩ n lim nlϱ lim ϩ lim nlϱ nlϱ n 1 1ϩ0 Here we used Equation with r EXAMPLE Calculate lim ■ SECTION 8.1 SEQUENCES ■ 415 SOLUTION Notice that both numerator and denominator approach infinity as n l ϱ We can’t apply l’Hospital’s Rule directly because it applies not to sequences but to functions of a real variable However, we can apply l’Hospital’s Rule to the related function f ͑x͒ ͑ln x͒͞x and obtain ln x 1͞x lim 0 xlϱ x lim xlϱ Therefore, by Theorem we have lim nlϱ an ln n 0 n ■ EXAMPLE Determine whether the sequence a n ͑Ϫ1͒ n is convergent or divergent SOLUTION If we write out the terms of the sequence, we obtain ͕Ϫ1, 1, Ϫ1, 1, Ϫ1, 1, Ϫ1, ͖ n _1 The graph of this sequence is shown in Figure Since the terms oscillate between and Ϫ1 infinitely often, a n does not approach any number Thus, lim n l ϱ ͑Ϫ1͒ n does not exist; that is, the sequence ͕͑Ϫ1͒ n ͖ is divergent ■ FIGURE EXAMPLE Evaluate lim The graph of the sequence in Example is shown in Figure and supports the answer ■ nlϱ ͑Ϫ1͒ n if it exists n SOLUTION an lim nlϱ Ϳ ͑Ϫ1͒ n n Ϳ lim nlϱ 0 n Therefore, by Theorem 6, lim n nlϱ ͑Ϫ1͒ n 0 n ■ Discuss the convergence of the sequence a n n!͞n n, where n! ؒ ؒ ؒ и и и ؒ n V EXAMPLE _1 FIGURE SOLUTION Both numerator and denominator approach infinity as n l ϱ but here we have no corresponding function for use with l’Hospital’s Rule (x! is not defined when x is not an integer) Let’s write out a few terms to get a feeling for what happens to a n as n gets large: a1 a2 an 1ؒ2 2ؒ2 a3 1ؒ2ؒ3 3ؒ3ؒ3 ؒ ؒ ؒ иии ؒ n n ؒ n ؒ n ؒ иии ؒ n It appears from these expressions and the graph in Figure 10 that the terms are decreasing and perhaps approach To confirm this, observe from Equation that an n ͩ ؒ ؒ иии ؒ n n ؒ n ؒ иии ؒ n ͪ 416 ■ CHAPTER SERIES CREATING GRAPHS OF SEQUENCES Some computer algebra systems have special commands that enable us to create sequences and graph them directly With most graphing calculators, however, sequences can be graphed by using parametric equations For instance, the sequence in Example can be graphed by entering the parametric equations ■ xt y t!͞t t Notice that the expression in parentheses is at most because the numerator is less than (or equal to) the denominator So Ͻ an ഛ We know that 1͞n l as n l ϱ Therefore, a n l as n l ϱ by the Squeeze Theorem V EXAMPLE and graphing in dot mode starting with t 1, setting the t -step equal to The result is shown in Figure 10 1 n ■ For what values of r is the sequence ͕r n ͖ convergent? SOLUTION We know from Section 1.6 and the graphs of the exponential functions in Section 3.1 that lim x l ϱ a x ϱ for a Ͼ and lim x l ϱ a x for Ͻ a Ͻ Therefore, putting a r and using Theorem 3, we have lim r n nlϱ ͭ ϱ if r Ͼ if Ͻ r Ͻ For the cases r and r we have 10 lim 1n lim nlϱ lim n lim and nlϱ nlϱ nlϱ FIGURE 10 Խ Խ If Ϫ1 Ͻ r Ͻ 0, then Ͻ r Ͻ 1, so Խ Խ Խ Խ lim r n lim r nlϱ nlϱ n 0 and therefore lim n l ϱ r n by Theorem If r ഛ Ϫ1, then ͕r n ͖ diverges as in Example Figure 11 shows the graphs for various values of r (The case r Ϫ1 is shown in Figure 8.) an an r>1 1 _1