lec 13 - Thursday, January 26 Lecture 13 : Infinite...

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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 well-defined 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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lec 13 - Thursday, January 26 Lecture 13 : Infinite...

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