Sequences PDF

CHAPTER IV SEQUENCES Sequences A sequence is a list or collection of numbers that is generated by some defining rule. Is nothing more than a list of numbers written in a specific order. Sequences are, basically, countably many ( – or higher-dimensional) vector...

CHAPTER IV SEQUENCES Sequences A sequence is a list or collection of numbers that is generated by some defining rule. Is nothing more than a list of numbers written in a specific order. Sequences are, basically, countably many ( – or higher-dimensional) vectors arranged in an ordered set that may or may not exhibit certain patterns. ▪ A sequence is a function f : → defined as f ( n ) = xn , and is usually denoted x1 , x2 , x3 ,..., xn or simply xn. We call xn the nth term of the sequence or the value of the sequence at n. Since the focus is on INFINITE sequences, each term in the sequence will be followed by another term as illustrated. 1|Pr epar ed by: MSAlimondo 2|Pr epar ed by: MSAlimondo Example: Function Notation: PRACTICE EXERCISE: Generate the sequence given the following: 3|Pr epar ed by: MSAlimondo 4|Pr epar ed by: MSAlimondo LIMITS OF SEQUENCES The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. A limit is a mathematically precise way to talk about approaching a value without having to evaluate it directly. Not every sequence has this behavior: those that do are called convergent, while those that don't are called divergent. Limits describe the long-term behavior of a sequence and are thus very useful in bounding them. ▪ A real number L is the limit of the sequence zn if the numbers in the sequence become closer and closer to L and not to any other number. In general sense, the limit of a sequence is the VALUE that it APPPROACHES with arbitrary closeness. Example: 1 If zn = c for some constant c, then lim xn → c, and if xn = , then lim xn → 0. n→ n n→ When the limit of a sequence as n →  approaches a single value, we say the sequence converges. ▪ We say that a sequence xn CONVERGES if there exists x0  such that for every   0 ,there exists a positive integer N such that x0  ( x0 −  , x0 +  ) or xn − x0   , for all n  N. If such a number x0  exists then it is unique.In this case, we say that the sequence xn converges to x0 and is called the limit of the sequence xn.If x0 is the limit of xn , we write lim xn = z0. n → Examples: 1. Does the following sequence generated by the function f ( n ) converge? ( −1 ) n 1 a. f ( n) = 1 + n b. f ( n ) = 10 n 5|Pr epar ed by: MSAlimondo CONVERGENT SEQUENCES IN A METRIC SPACE In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis(is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions) and geometry( It is concerned with properties of space such as the distance, shape, size, and relative position of figures). Let X := ( X ,d ) denote a metric space. If the metric is not specified, we assume that the standard Euclidean metric is assigned. RECALL: a metric space is an ordered pair ( ,d) where  is a set and dis a metric on  , i.e., a function d :   →  SEQUENCE IN A METRIC SPACE ▪ A sequence in X is a function from → X by assigning a value f ( n )  X to each natural number n . The set ( xn ) n  = ( xn ) = xn n   =  x , x , x ,... is called a sequence of X.The 1 2 3 elements of ( xn ) n are called terms of the sequence. ▪ A sequence in a metric space X is a function x : N → X. In the usual notation for functions the value of the function x at the integer n is written x ( n ). In this chapter, sequences we will be denoted by xn instead of x ( n ). ▪ For any sequence xn we can consider the set of values it attains, namely      xn n   =  y y  xn for some n       ▪ ALTERNATE: A sequence in a metric space ( X ,d ) is a function x : → X. In previous lessons of sequences, the nth element in the sequence is written xn      and use the notation  xn  , or more precisely,  xn      n=1 6|Pr epar ed by: MSAlimondo  Example: 1 1 1 1  1 1. The sequence  , , , ,... can be written as   n  and is nothing 1 2 3 4   n 1 but a function g : → defined by g ( n ) =. n 1 Graph of g ( n ) = n  1 , 2 , 3 , 4 ,..., 49 n 2. Consider the sequence defined by ( −1 )  n n 0. a. List the first five terms of the sequence.____________________________________ b. Range:__________________________   3. Consider the sequence  n .   n a. Describe the function/rule. b. Write the first seven terms of the sequence. c. Range:  n−1  1 2 3  4. Consider   := 0 , , , ,...  n  n  2 3 4  a. Rewrite the function. b. Plot the sequence for n  1 , 2 , 3 , 4 ,..., 49 7|Pr epar ed by: MSAlimondo NOTE: It is important to distinguish set from the sequence itself. Example: If X = and xn = 1 for all n  , then the sequence xn is 1,1,1,…,i.e. an infinite sequence of ones.The set of values of this sequence is 1 which is a subset of , a singleton. BOUNDED SEQUENCES   ▪ A sequence  xn  is bounded if there exists a point p  X and B  such     that d ( p, xn )  B for all n . In other words, a sequence  xn  is bounded     whenever the set  xn : n   is bounded.   SUBSEQUENCE ▪ A subsequence of a given sequence xn is any other sequence yk that is of the form yk = xnk where nk is an increasing sequence of natural numbers, i.e. n1  n2  n3...    ▪ ALTERNATE: If  n j  is a sequence of natural numbers such that n j +1  n j   n=1      for all j then the sequence  xn j  is said to be a subsequence of  xn .   n =1   Example: 1. xn = ( −1 ) , so xn = −1 ,1 , −1 ,1 ,..., then yk = x2k is the subsequence that selects n all even numbered entries in the sequence. Thus yk = 1 ,1 ,1 ,1 ,1 ,.... It will often be useful to realize that in an increasing sequence of natural numbers nk one always has nk  k. This is true or holds since the set 8|Pr epar ed by: MSAlimondo    n1 ,n2 , n3 ,...nk −1 ,nk  consists of k different natural numbers, all of which are less   than or equal to nk.This can only happen if nk  k. CONVERGENT SEQUENCES   ▪ A sequence  xn  in a metric space ( X ,d ) is said to converge to a point p  X   , if for   0 ,there exists an M  such that d ( xn , p ) for all point n  M.   The point p is said to be the limit of  xn . We write lim xn := p.  n → ▪ A sequence that converges is said to be convergent. Otherwise, the sequence is said to be divergent. ▪ A convergent sequence in a metric space has a unique limit. 9|Pr epar ed by: MSAlimondo CAUCHY SEQUENCE One of the problems with deciding if a sequence is convergent is that you need to have a limit before you can test the definition. In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. efinition 10 | P r e p a r e d b y : M S A l i m o n d o  1.. Any convergent sequence is bounded (both above and below) Arithmetic properties of Cauchy Sequence 11 | P r e p a r e d b y : M S A l i m o n d o 1.1. Cauchy Sequences Complete Metric Space 12 | P r e p a r e d b y : M S A l i m o n d o. 13 | P r e p a r e d b y : M S A l i m o n d o