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Unformatted text preview: Thursday, January 26 Lecture 13 : Infinite sequences of numbers . (Refers to Section 8.1 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Find the limit of simple sequences. 13.1 Definition A sequence of numbers is a function a ( i ) = a i whose domain is a set of ordered integers. Most often, the domain is {1, 2, 3,... } 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. 13.1.1 Example The ordered set { a i : i = 1, 2, 3,...} where a i = 2 i + 3 is a welldefined sequence. Essentially it is the function f ( x ) = 2 x + 3 with the positive integers as domain. 13.1.2 Definition A recursive sequence is a sequence whose i th term is expressed in terms of a formula involving previous terms of the sequence. 13.1.2.1 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 sometimes difficult 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, This closed form describes the n th Fibonacci number given that the first two terms are a (0) = 0 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 term....
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 Winter '08
 ZHOU
 Calculus

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