sequences - Innite Sequences and Series Dr. Philippe B....

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Infinite Sequences and Series Dr. Philippe B. Laval Kennesaw State University Abstract This hand out is an introduction to sequences and series. It gives the main definitions and presents some elementary results. 1 Sequences and Series - General Concepts (8.1 &8 .2) 1.1 Infinite Sequences 1.1.1 Elementary Concepts A sequence is simply a list of numbers, written in a definite order: a 1 ,a 2 3 ,...a n n +1 ,... where the elements a i represent numbers. In this section, we only concentrate on infinite sequences. So, every number a n in the sequence has a successor a n +1 ; the sequence never stops. The index of the numbers does not have to start at 1 , though it does most of the time. If we call n 0 the value of the starting index, then there is a number a n for every n n 0 . Thus, we can define a function f such that a n = f ( n ) . In this class, we will concentrate on sequences of real numbers. A more formal definition of a sequence is as follows: Definition 1 (sequence) A sequence of real numbers is a real-valued function f whose domain is a subset of the non-negative integers, that is a set of the form { n 0 ,n 0 +1 } where n 0 is an integer such that n 0 0 .. The numbers a n = f ( n ) are called the terms of the sequence. The typical notation for a sequence is ( a n ) ,o r { a n } or { a n } n =1 where a n denotes the general term of the sequence. Remark When a sequence is given by a function { f ( n ) } n = n 0 , the function f must be defined for every n n 0 . A sequence can be given different ways 1
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The elements of the sequence are given. A formula to generate the terms of the sequence is given. A recursive formula to generate the terms of the sequence is given. Example 2 The examples below illustrate sequences given by listing all the el- ements, 1. { 1 , 5 , 10 , 4 , 98 , 1000 , 0 , 2 ,... } . 2. ± 1 , 1 2 , 1 3 ² . Though the elements are listed, we can also guess a formula to generate them, what is it? 3. {− 1 , 1 , 1 , 1 , 1 , 1 } . Though the elements are listed, we can also guess a formula to generate them, what is it? Example 3 The examples below illustrate sequences given by a simple formula. Notice that n does not have to start at 1 . 1. ± 1 n ² n =1 . The elements of the sequence are: ± 1 , 1 2 , 1 3 ² . The general term of the sequence is a n = 1 n . 2. ³ n 3 ´ n =3 . The elements are ³ 0 , 1 , 2 , 3 ´ . In this case, n could not start at 1 . 3. { ( 1) n } n =2 . The elements of the sequence are { 1 , 1 , 1 , 1 } . Example 4 The examples below illustrate sequences given by a recursive for- mula. 1. ± a 1 =1 a n =2 a n 1 +5 . We can use this formula to generate all the terms. But the terms have to be generated in order. For example, in order to get a 10 ,weneedtoknow a 9 , and so on. Using the formula, we get that a 2 a 1 =7 Having found a 2 , we can now generate a 3 a 3 a 2 9 Andsoon .
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sequences - Innite Sequences and Series Dr. Philippe B....

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