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Sequences Sequences and Series Sequences The Pinching Theorem: This tool gives us a means for comparing a new, perhaps difficult and mysterious, sequence with other more familiar sequences. Theorem: Suppose that, aj,bj, and cj are sequences. Sequences a , b j j 1 j j 1 a j bj c j , and c j j 1 for every j if lim j a j lim j c j l then lim j b j l Sequences Infinite Sequences of Real Numbers a1 , a 2 , a 3 or a j j 1 Sequences Example: Discuss the sequence 2, 1, 4, 3, 6, 5,…. We see that a1 2, a2 1, a3 4, a4 3, a5 6, a6 5,.... or f (1) 2, f (2) 1, f (3) 4, f (4) 5, f (5) 6 f (6) 5,..... Sequences Remark: It would be a mistake to think that every sequence is given by a rule. Far from it. But many sequences do come from rules, and it is always interesting to determine what that rule is. Sequences Example: How does the sequence 1,2,3,4,5,… Differ from the sequence in previous example. This new sequence has the same values as the sequence in last example. But they occur in a different order. Since a sequence is by definition an ordered list. Sequences Occasionally it will prove useful to begin a sequence with an index different from 1. An example is 3 j 5 j 4 7, 10, 13, 16, Sequences Example: Discuss whether the sequence tends to a limiting value. j a j 2 , or We write 1 1 1 , , , 2 4 8 2 j j 1 Sequences Then we see that the terms become (and remain) arbitrarily close to zero. It seems plausible to say that the sequence tends to zero. Sequences Example: Does the sequence tend to a limit? a j ( 1) j we write 1, 1, 1, 1, The sequence does not seem to tend to any limit. The sequence does not become and remain close to a single value. Therefore we say that it has no limit. Sequences Example: Does the sequence tend to a limit? aj j 3 We write 1,8,27,64,... Sequences Example: A quantity of radioactive material decays. At the beginning of each week there is half as much as there was the previous week. The initial quantity is 5 grams. Use sequence notation to express the amount of material at the beginning of the jth week. Sequences Solution: The amount of radioactive material at the first week is 5, second week is 5/2, The third week is 5/4, and so on, The amount of radioactive material tends to 0 as time tends to ∞ 1 a j 5 2 j 1 5 5 5 or 5, , , , 2 4 8 Sequences Example: Discuss convergence for the sequence 1, ½, 1/3, ¼,…. We conclude that the sequence converges. Sequences Example: Find the limit of the sequence a j 2 j j 1 or a j 2 2 j 1 or a j 2 1 1/ j Sequences Example: Find the limit of the sequence 10 j 8 j aj 10 j or 8 a j 1 10 j Sequences Sequences with and without patterns: Sometimes a sequence will come from an obvious pattern or rule, and sometimes not. Example: What is the next element of the sequence 6, 6, 1, 7, 10, 2, 5, 3, 2, 5, 3 Sequences Example: Find the pattern in the sequence and find the limit: 1 4 3 6 5 , , , , 2 3 4 5 6 The sequence is rewritten as 2, 1 1 1 1 1 ,1 ,1 ,1 ,1 , 1 2 3 4 5 1 1 1 1 1 1 1 1 . ,1 ( 1) 2 1 . ,1 ( 1) 3 1 . ,1 ( 1) 4 1 . , 1 2 3 4 the j th term 1 a j 1 ( 1) j 1 . Limit j a j 1 1 j Sequences Example: Evaluate lim j (sin j ) 2 j (sin j ) 2 1 let b j then 0 bj j j 1 a j 0, cj for every j j We observe lim j a j lim j c j 0 lim j (sin j ) 2 lim j b j 0 j Sequences Some Special Sequences: Theorem let S be a real number and the sequence is: j s j1 If s<0 then the sequence converges to 0 If s>0 then the sequence diverges If s=0 then the sequence is just constant sequence 1,1,1,…, and converges to 1. Sequences Theorem let t be a real number and the sequence is: t j j1 If |t|<1 then the sequence converges to 0 If |t|>1 then the sequence diverges If t=1 then the sequence is the constant sequence 1,1,1,1,…which converges to 1 If t=-1 then the sequence diverges. Sequences Examples: lim j lim j lim j j 2j 2 4j 3/ 2 2 3 j 4 j.sin(1 / j ) 2 j j lim j j 2 (1 cos(1 / j ))