This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.
 Fall '10
 LUXNU
 Math

Click to edit the document details