sec11_1 - 11.1 Sequences 1 Sequences Denition 1.1 A...

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11.1 Sequences 1. Sequences Definition 1.1. A sequence is a function whose domain is the set of positive integers. Example 1.1. f ( n ) = n 2 where n = 1 , 2 , 3 , . . . Notation 1.1. The sequence f ( n ) = a n may be written in a variety of ways. Some examples are { a n } n =1 , { a n } , { a 1 , a 2 , a 3 , . . . } , ( a 1 , a 2 , a 3 , . . . ) , a n Example 1.2. f ( n ) = n 2 where n = 1 , 2 , 3 , . . . could be written: Example 1.3. (problem 10) Find a formula for the general term a n of the sequences, assuming that the pattern of the first few terms continues. { 1 , 1 3 , 1 9 , 1 27 , 1 81 , ... } Example 1.4. (problem 12) {- 1 4 , 2 9 , - 3 16 , 4 25 , ... } 1
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Section 11.1 Sequences 2 2. Recursion Definition 2.1. A recursive sequence uses previous terms to define later terms. Example 2.1. Find the first four terms of the following sequence. (This sequence is called the Fibonacci Sequence) a 1 = 1 , a 2 = 1 , a n = a n - 1 + a n - 2 for n 3 3. Convergence and Divergence Definition 3.1. A sequence { a n } has the limit L and we write lim n →∞ a n = L if we can make the terms a n as close to L as we like by taking n sufficiently large. If lim n →∞ a n
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