lect138_14_w11 - Friday February 4 Lecture 14 : Infinite...

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Friday February 4 Lecture 14 : Infinite sequences of numbers . (Refers to Section 11.1 in your text) Expectations: 1. Find the limit of simple sequences. 14.1 Definition A sequence of numbers is a function a ( i )= a i whose domain is the set of positive integers. It can be denoted as { a i : i = 1, 2, 3,. ..} or { a 1 , a 2 , a 3 , . ..}, or simply by a 1 , a 2 , a 3 , . ... We say that the elements of the sequence are indexed with the natural numbers. Its elements are referred to as the terms of the sequence , where a 1 is the first term, a 2 the second, and so on. - The order must be respected. Altering the order may alter some of the convergence properties of the sequence, as we will see later. 14.1.1 Example The ordered set { a i : i = 1, 2, 3,. ..} where a i = 2 i + 3 is a well-defined sequence. Essentially it is the function f ( x ) = 2 x + 3 with the positive integers as domain. 14.1.2 Definition A recursive sequence is a sequence whose i th term is expressed in terms of a formula involving previous terms of the sequence. Example Consider the sequence { a i } defined as This recursive sequence{ a i } is called a Fibonacci sequence . Some recursive sequences have a " closed " form. It is not always easy to determine the closed form of a recursive sequence, or even determine if it has one. The above Fibonacci sequence can be shown to have the closed form,
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This closed form describes the n th Fibonacci number given that the first two terms are a (0) = 1 and a (1) = 1. (We can prove this formula by using linear algebra) It is useful since we are not required to compute the previous terms to obtain the n th
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This note was uploaded on 11/10/2011 for the course MATH 138 taught by Professor Anoymous during the Spring '07 term at Waterloo.

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lect138_14_w11 - Friday February 4 Lecture 14 : Infinite...

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