chap10-et-instructor-solutions

Chap10-et-instructor - 10 INFINITE SERIES 10.1 Sequences Preliminary Questions 1 What is a4 for the sequence an = n 2 n SOLUTION Substituting n = 4

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10 INFINITE SERIES 10.1 Sequences Preliminary Questions 1. What is a 4 for the sequence a n = n 2 n ? SOLUTION Substituting n = 4 in the expression for a n gives a 4 = 4 2 4 = 12 . 2. Which of the following sequences converge to zero? (a) n 2 n 2 + 1 (b) 2 n (c) µ 1 2 n (a) This sequence does not converge to zero: lim n →∞ n 2 n 2 + 1 = lim x →∞ x 2 x 2 + 1 = lim x →∞ 1 1 + 1 x 2 = 1 1 + 0 = 1 . (b) This sequence does not converge to zero: this is a geometric sequence with r = 2 > 1; hence, the sequence diverges to . (c) Recall that if | a n | converges to 0, then a n must also converge to zero. Here, ¯ ¯ ¯ ¯ µ 1 2 n ¯ ¯ ¯ ¯ = µ 1 2 n , which is a geometric sequence with 0 < r < 1; hence, ( 1 2 ) n converges to zero. It therefore follows that ( 1 2 ) n converges to zero. 3. Let a n be the n th decimal approximation to 2. That is, a 1 = 1, a 2 = 1 . 4, a 3 = 1 . 41, etc. What is lim n →∞ a n ? lim n →∞ a n = 2. 4. Which sequence is deFned recursively? (a) a n = p 2 + n 1 (b) b n = p 4 + b n 1 (a) a n can be computed directly, since it depends on n only and not on preceding terms. Therefore a n is deFned explicitly and not recursively. (b) b n is computed in terms of the preceding term b n 1 , hence the sequence { b n } is deFned recursively. 5. Theorem 5 says that every convergent sequence is bounded. Which of the following statements follow from Theorem 5 and which are false? If false, give a counterexample. (a) If { a n } is bounded, then it converges. (b) If { a n } is not bounded, then it diverges. (c) If { a n } diverges, then it is not bounded. (a) This statement is false. The sequence a n = cos π n is bounded since 1 cos n 1fo ra l l n , but it does not converge: since a n = cos n = ( 1 ) n , the terms assume the two values 1 and 1 alternately, hence they do not approach one value. (b) By Theorem 5, a converging sequence must be bounded. Therefore, if a sequence is not bounded, it certainly does not converge. (c) The statement is false. The sequence a n = ( 1 ) n is bounded, but it does not approach one limit.
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SECTION 10.1 Sequences 1133 Exercises 1. Match the sequence with the general term: a 1 , a 2 , a 3 , a 4 ,... General term (a) 1 2 , 2 3 , 3 4 , 4 5 (i) cos π n (b) 1 , 1 , 1 , 1 (ii) n ! 2 n (c) 1 , 1 , 1 , 1 (iii) ( 1 ) n + 1 (d) 1 2 , 2 4 , 6 8 , 24 16 ... (iv) n n + 1 SOLUTION (a) The numerator of each term is the same as the index of the term, and the denominator is one more than the numerator; hence a n = n n + 1 , n = 1 , 2 , 3 ,... . (b) The terms of this sequence are alternating between 1 and 1 so that the positive terms are in the even places. Since cos n = 1foreven n and cos n =− 1 for odd n ,wehave a n = cos n , n = 1 , 2 (c) The terms a n are 1 for odd n and n . Hence, a n = ( 1 ) n + 1 , n = 1 , 2 (d) The numerator of each term is n ! , and the denominator is 2 n ; hence, a n = n ! 2 n , n = 1 , 2 , 3 2. Let a n = 1 2 n 1 for n = 1 , 2 , 3 ,.... Write out the Frst three terms of the following sequences.
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This note was uploaded on 11/17/2009 for the course MATH 114 taught by Professor Carnegie during the Spring '09 term at Carleton.

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Chap10-et-instructor - 10 INFINITE SERIES 10.1 Sequences Preliminary Questions 1 What is a4 for the sequence an = n 2 n SOLUTION Substituting n = 4

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