note13 - Math021, week 13 Sequences Definition 13.1 An...

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Unformatted text preview: Math021, week 13 Sequences Definition 13.1 An infinite sequence of numbers (or just a sequence) a 1 ,a 2 ,a 3 ,... is usually denoted by the symbol { a n } . Example 13.2 Let a and d be numbers. A sequence { a n } defined by a n = a + ( n- 1) d for all n is usually called an arithematic progression (AP). Example 13.3 Let a and q be numbers. A sequence { a } defined by a n = aq n- 1 for all n is usually called a geometric progression (GP). Definition 13.4 Let { a n } be a sequence and L be a number. We say that { a n } converges to L if a n can be as close to L as one wish provided that n is sufficiently large. In symbol, lim n →∞ a n = L. We say that { a n } converges if it converges to a certain number L . Otherwise, we say that { a n } diverges. Definition 13.5 A sequence { a n } diverges to positive (negative) infinity if a n (- a n ) can be as large as one wish provided that n is sufficiently large. Example 13.6 Evaluate lim n →∞ 1 n if it converges. solution: Since 1 n can be arbitrarily small provided that n is large enough. { 1 n } converges and lim n →∞ 1 n = 0 . 1 Example 13.7 Evaluate lim n →∞ (- 1) n if it converges. solution: Since (- 1) n is 1 when n is even and is- 1 when n is odd. (- 1) n does not approach any number when n is large. Thus, the sequence { (- 1) n } diverges. Example 13.8 Determine if the arithmetic progression { a +( n- 1) d } converges. solution: Case 1: d = 0, the given sequence is constantly a . Hence it converges to a . Case 2: d > 0, the given sequence has a positive number d added to each term when com- pared to the previous term. Thus, this sequence diverges to + ∞ . Case 3: d < 0, Similarly, the given sequence diverges to-∞ in this case. Example 13.9 Determine if the geometric progression { aq n- 1 } converges. solution: Case 1: | q | < 1, aq n- 1 is as small as one wish provided that n is sufficiently large. Thus, the given sequence converges to 0. Case 2: | q | > 1, | aq n- 1 | is as large as one wish provided that n is sufficiently large. Thus, the given sequence diverges. Case 3: q = 1, The given sequence is constantly a , it converges to a . Case 4: q =- 1, The given sequence is alternating between a and- a . It diverges when a = 0, and converges when a 6 = 0. Theorem 13.10 If { a n } and { b n } are sequences, 1. lim n →∞ ( a n + b n ) = lim n →∞ a n + lim n →∞ b n ....
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This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.

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note13 - Math021, week 13 Sequences Definition 13.1 An...

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