sec11_1

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

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11.1 Sequences 1. Sequences Defnition 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 3 } , ( a 1 2 3 ) n Example 1.2. f ( n n 2 where n , 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 ±rst 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 Defnition 2.1. A recursive sequence uses previous terms to defne later terms. Example 2.1. Find the frst ±our terms o± the ±ollowing sequence. (This sequence is called the Fibonacci Sequence) a 1 =1 ,a 2 n = a n - 1 + a n - 2 ±or n 3 3. Convergence and Divergence Defnition 3.1. A sequence { a n } has the limit L and we write lim n →∞ a n = L i± we can make the terms a n as close to L as we like by taking n sufficiently large. I± lim n →∞ a n exists, we say the sequence
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This note was uploaded on 03/14/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

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

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