Sequences

11 Key excerpts on "Sequences"

• 

  eBook - ePub

  Mathematics for Engineers and Scientists

  • Alan Jeffrey(Author)
  • 2004(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  + … is a series.

  If a sequence is composed of elements or terms u belonging to some set S , then it is conventional to indicate their order by adding a numerical suffix to each term. Consecutive terms in the sequence are usually numbered sequentially, starting from unity, so that the first few terms of a sequence involving u would be denoted by u 1 , u 2 , u 3 ,… . Rather than write out a number of terms in this manner this sequence is often represented by {

  un

  }, where

  un

  is the n th term, or general term, of the sequence. The sequence depends on the set chosen for S and the way suffixes are allocated to elements of S . A sequence will be said to be infinite or finite according to whether the number of terms it contains is infinite or finite and, unless explicitly stated, all Sequences will be assumed to be infinite. The notation for a sequence is often modified to

  {

  u n

  }

  n = 1

  N

  when only a finite number N of terms are involved, so that

  {

  u n

  }

  n = 1

  N

  =

  u 1

  ,

  u 2

  , … ,

  u N

  .

  As an example of an infinite numerical sequence, let S be the set of real numbers and the rule by which suffixes are allocated be that to each integer suffix n we allocate the number 1/2n which belongs to R . We thus arrive at the finite sequence u 1 = 1/2, u 2 = 1/22 , u 3 = 1/23

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  Proofs and Ideas

  A Prelude to Advanced Mathematics

  • B. Sethuraman(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  11 Sequences, Series, Continuity, Limits If you have had a calculus course before, you would likely have seen the notions of Sequences and series, continuity of functions, and limits of functions, if only informally. We will study these notions in slightly greater depth and with slightly greater rigor here than is typical in a first (computationally oriented) calculus course. As we will see, the last two notions of continuity of functions and limits of functions, which are closely related, are easiest to understand via the notion of convergent Sequences (although there are alternative approaches as well). Accordingly, we will begin the chapter with Sequences. Sequences form an integral part of the foundations of calculus: they allow the mi-croscopic study of functions (from ℝ to ℝ , but more generally, also from ℂ to ℂ ). They enable us to approximate the continuous (and uncountable) aspect of functions de-fined on ℝ by discrete behaviors and thus allow us to slow down such functions to a human scale. But more is true: Sequences do not merely serve as approximations of functions from ℝ to ℝ , they actually enable us to define attributes of functions in terms of attributes of Sequences obtained from them. A good understanding of Sequences is therefore vital to the further study of mathematics. Especially vital is the notion of convergence of Sequences: we will focus on a rigorous definition of this notion, and on rigorous proofs that various Sequences converge. In the later portion of this chapter, we will study how Sequences can be used to pin down the notion of continuity of real-valued functions on subsets of the real numbers, more precisely, functions ?∶ ? → ℝ , where ? is an open interval in ℝ . The more in-tuitive definition of continuity is expressed via Sequences, although there is a different definition that is easier to work with when showing that specific functions are contin-uous.

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  Sequences and Series in Calculus

  • Joseph D. Fehribach(Author)
  • 2023(Publication Date)
  • De Gruyter

    (Publisher)

  1  Sequences in ℝ

  Our discussion begins with a very basic concept in mathematics: Sequences. Probably most people have at least a general intuitive idea of what a sequence is. One simple example of a sequence is

  { 1 , 2 , 3 , 4 , 5 , … } ,

  while another more irregular sequence is

  { 3 , 7 , π , 2 / 5 , − 6 , … } .

  In the second case, it is not possible to predict what the next entry will be after −6, but nonetheless, both examples seem to satisfy the essential sense of what a sequence is. What is needed now is an exact mathematical definition:

  Definition.

  A sequence is a function

  a : A → R

  where either

  A =

  Z +

  : = { 1 , 2 , 3 , … }

  or

  A = N : = { 0 , 1 , 2 , 3 , … }

  (the natural numbers) or perhaps A is any countable set.1 Thus, either

  a : {

  a n

  } = {

  a 1

  ,

  a 2

  ,

  a 3

  , … }

  or

  a : {

  a n

  } = {

  a 0

  ,

  a 1

  ,

  a 2

  ,

  a 3

  , … } ,

  where in both cases

  a n

  ∈ R

  . We refer to

  a n

  as the n-th element or entry of the sequence.

  Remarks.

  1.

  Notice that a sequence differs from a set in that a sequence has an order, whereas a set is simply a collection of elements. Technically, there is no first element in a set, but there is in a sequence.

  2.

  This is not the most general definition of the word “sequence,” but it is easily general enough for our purposes. It contains all the key ideas to discuss Sequences. In particular, we will not discuss finite Sequences.

  3.

  Notice that for our purposes, Sequences begin either with

  n = 0

  or with

  n = 1

  . Beginning with a zeroth element is less common, but it occurs sometimes when a zeroth element makes sense. Among other places, this happens in computer science. Still, unless there is some specific reason to do otherwise, let

  A =

  Z +

  .

  A sequence can be depicted graphically on the real line as in Figure 1.1 .

  Figure 1.1

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  Learning and Teaching Mathematics using Simulations

  Plus 2000 Examples from Physics

  • Dieter Röss(Author)
  • 2011(Publication Date)
  • De Gruyter

    (Publisher)

  4 Sequences of numbers and series 4.1 Sequences and series By Sequence repeated application of the same arithmetic operations on an initial number A , one creates a logically connected sequence of numbers, which show interesting properties (to guess the formation law of a sequence and thus to continue the initial numbers of a given sequence is a popular type of puzzle). In the following the letters m; n; i; j are used to indicate the position of terms in Sequences. They can be 0 or positive integers. If there is no upper limit for the number of terms in a sequence or for the terms in a series ( m ! 1 ), we refer to an infinite sequence or series. 4.1.1 Sequence and series of the natural numbers The particularly simple arithmetic sequence of the natural numbers is created via the repeated addition of the unit 1 ; the individual term is characterized by the lower index ( 1; 2; : : : ), which itself is an increasing natural number. A 1 D 1 I A n C 1 D A n C 1 for n 1 ! A n D 1; 2; 3; 4; 5; 6; : : : : We now define the difference quotient for the terms of an arbitrary sequence with different indices i and j . This number is a measure for the change between two terms with different indices and thus for the growth of the sequence in the interval given by the indices: ĩA i;j D A i A j I ĩ i;j D i j difference quotient: ĩA i;j ĩ i;j D A i A j i j : For consecutive terms, the index interval is 1 and the difference quotient is equal to the difference between the terms: ĩA i;i 1 D A i A i 1 I ĩ i;i 1 D i .i 1/ D 1 difference quotient: ĩA i;i 1 ĩ i;i 1 D A i A i 1 : 4.1 Sequences and series 36 For the sequence of the natural numbers, the difference between consecutive terms is constant and equal to 1 . Therefore their difference quotient is also constant and equal to 1 . ĩA i;i 1 D A i A i 1 D 1 ! difference quotient D 1: The arithmetic sequence has constant growth of consecutive terms.

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  Functions Modeling Change

  A Preparation for Calculus

  • Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Any ordered list of numbers, such as 1792, 1796, 1800, 1804, … is called a sequence, and the individual numbers are the terms of the sequence. A sequence can be a finite list, such as the sequence of past presidential election years, or it can be an infinite list, as in parts (a)–(c) of Example 1. Example 1 (a) 0, 1, 4, 9, 16, 25, … is the infinite sequence of squares of integers. (b) 2, 4, 8, 16, 32, … is the infinite sequence of positive integer powers of 2. (c) 3, 1, 4, 1, 5, 9, … is the infinite sequence of digits in the decimal expansion of . (d) 3.9, 5.3, 7.2, 9.6, 12.9, 17.1, 23.1, 38.6, 50.2 is the finite sequence of US population figures, in millions, for the first 10 census reports (1790 to 1880). (e) 92.8, 122, 144.5, 171, 189, 207.1 is the finite sequence of smartphone users in the US, 1 in mil- lions, from the years 2011 to 2016. Notation for Sequences We denote the terms of a sequence by  1 ,  2 ,  3 , … ,   , … so that  1 is the first term,  2 is the second term, and so on. We use   to denote the  th or general term of the sequence. If there is a pattern in the sequence, we may be able to find a formula for   . Example 2 Find the first three terms and the 98 th term of the sequence. (a)   = 1 + √  (b)   = (−1)    + 1 Solution (a)  1 = 1 + √ 1 = 2,  2 = 1 + √ 2 ≈ 2.414,  3 = 1 + √ 3 ≈ 2.732, and  98 = 1 + √ 98 ≈ 10.899. (b)  1 = (−1) 1 1 1 + 1 =− 1 2 ,  2 = (−1) 2 2 2 + 1 = 2 3 ,  3 = (−1) 3 3 3 + 1 =− 3 4 , and  98 = (−1) 98 98 98 + 1 = 98 99 . This sequence is called alternating because the terms alternate in sign. A sequence can be thought of as a function whose domain is a set of integers. Each term of the sequence is an output value for the function, so   =  (). Example 3 List the first five terms of the sequence   =  (), where  () = 500 − 10. Solution Evaluate  () for  = 1, 2, 3, 4, 5:  1 =  (1) = 500 − 10 ⋅ 1 = 490 and  2 =  (2) = 500 − 10 ⋅ 2 = 480.

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  Precalculus

  Functions and Graphs

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  660 CHAPTER 9 Sequences, Series, and Probability Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. In general, given any infinite sequence, a 1 , a 2 , . . . , a n , . . . , the expression a 1 1 a 2 1 ? ? ? 1 a n 1 ? ? ? is called an infinite series or simply a series. We denote this series by O ` n 5 1 a n . Each number a k is a term of the series, and a n is the n th term. Since only finite sums may be added algebraically, it is necessary to define what is meant by an infinite sum. Consider the sequence of partial sums S 1 , S 2 , . . . , S n , . . . . If there is a number S such that S n l S as n l ` , then, as in our discussion of infinite geometric series, S is the sum of the infinite series and we write S 5 a 1 1 a 2 1 ? ? ? 1 a n 1 ? ? ? . In the previous example we found that the infinite repeating decimal 5.4272727. . . corresponds to the rational number 597 110 . Since 597 110 is the sum of an infinite series determined by the decimal, we may write 597 110 5 5.4 1 0.027 1 0.00027 1 0.0000027 1 ? ? ? . If the terms of an infinite sequence are alternately positive and negative, as in the expression a 1 1 s 2 a 2 d 1 a 3 1 s 2 a 4 d 1 ? ? ? 1 fs 2 1 d n 1 1 a n g 1 ? ? ? for positive real numbers a k , then the expression is an alternating infinite series and we write it in the form a 1 2 a 2 1 a 3 2 a 4 1 ? ? ? 1 s 2 1 d n 1 1 a n 1 ? ? ? . The most common types of alternating infinite series are infinite geometric series in which the common ratio r is negative.

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  A Concrete Introduction to Real Analysis

  • Robert Carlson(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  a discussion of infinite Sequences is needed. Until the description is completed a number will be an element of an Archimedean ordered field.

  Intuitively, an infinite sequence is simply an infinite list of numbers. The k–th term of the sequence is denoted ck or c(k). Examples include the Sequences

  1 ,

  1 2

  ,

  1 3

  ,

  1 4

  , …  

  c k

    =  

  1 k

  ,   k   =   1 , 2 , 3 , … ,

  1 ,   − 1 , 1 , − 1 , … ,  

  c k

    =  

  (

  − 1

  )

  k

  ,   k   =   0 , 1 , 2 , … ,

  and

  3,3 .1, 3 .14,3 .141, 3 .1415, … ,

  where ck is the first k digits of the decimal expansion of π.

  In the usual mathematical language, an infinite sequence, or simply a sequence, is a function c whose domain is the set ℕ of positive integers 1, 2, 3,…. The value c(k), or more commonly ck , of the function at k is called the k – th term of the sequence. For our purposes the values c(k) will typically be numbers, although the idea extends to more complex objects. A slight extension of the idea allows the domain to be the set of nonnegative integers.

  Although a sequence is a function, it is common to use a special notation for Sequences. As noted above, the terms are often written ck instead of c(k). The sequence itself is denoted {ck }. As an abbreviation, people often write “the sequence ck ,” instead of “the sequence {ck },” although this can create some confusion between the entire sequence and its k-th term.

  The notion of a limit is the most important idea connected with Sequences. Say that the sequence of numbers {ck } has the number L as a limit if for any ϵ > 0 there is an integer N such that

  |

  c k

    −   L

  |

    <   ϵ ,   whenever   k   ≥   N .

  To emphasize the dependence of N on ϵ we may write Nϵ or N(ϵ). In mathematical shorthand the existence of a limit is written as

  lim

  k   →   ∞

   

  c k

    =   L .

  An equivalent statement is that the sequence {ck } converges to L. This definition has a graphical interpretation which illustrates the utility of the function interpretation of a sequence. The statement that the sequence has the limit L is the same as saying that the graph of the function c(k) has a horizontal asymptote y = L, as shown in Figure 2.1 (where L

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  An Introduction to Fourier Analysis

  • Russell L. Herman(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  We then call a n the n th term of the sequence. Furthermore, we will denote Sequences by { a n } ∞ n = 1 . Sometimes we will only give the n th term of the sequence and will assume that n ∈ N unless otherwise noted. a n n 1 2 3 4 5 6 7 8 9 10 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 Figure 1 . 2 : Plot of the terms of the se-quence a n = 1 2 n , for n = 1, 2, . . . , 10. Another way to define a particular sequence is recursively. A recursive sequence is defined in two steps: 1 . The value of first term (or first few terms) is given. 2 . A rule, or recursion formula, to determine later terms from earlier ones is given. Example 1 . 1 . A typical example is given by the Fibonacci 2 sequence. 2 Leonardo Pisano Fibonacci (c. 1170 -c. 1250 ) is best known for this sequence of numbers. This sequence is the solu-tion of a problem in one of his books: A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive http://www-history.mcs.st-and.ac.uk. It can be defined by the recursion formula a n + 1 = a n + a n -1 , n ≥ 2 and the starting values of a 1 = 0 and a 1 = 1. The resulting sequence is { a n } ∞ n = 1 = { 0, 1, 1, 2, 3, 5, 8, . . . } . Writing the general expression for the n th term is possible, but it is not as simply stated. Recursive defini-tions are often useful in doing computations for large values of n . 1 . 2 Convergence of Sequences a n n 1 2 3 4 5 6 7 8 9 10 .1 .2 .3 .4 .5 -.1 -.2 -.3 -.4 -.5 Figure 1 . 3 : Plot of the terms of the se-quence a n = ( -1 ) n 2 n , for n = 1, 2, . . . , 10. N ext we are interested in the behavior of Sequences as n gets large. For the sequence defined by a n = n -1, we find the behavior as shown in Figure 1 . 1 . Notice that as n gets large, a n also gets large.

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  Foundations of Mathematical Analysis

  • Richard Johnsonbaugh, W.E. Pfaffenberger(Authors)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  IV

  Sequences of Real Numbers

  Analysis is concerned in one form or another with limits. We begin our study of real analysis with the study of limits of real Sequences.

  10. Limit of a Sequence

  Let X be a set. A sequence of elements of X is a “list” of elements from the set X. In this chapter we will be concerned with Sequences of real numbers. We first make the notion of sequence precise.

  Definition 10.1 Let X be a set. A sequence of elements of X is a function from the set of positive integers into X.

  In particular, a real sequence (or sequence of real numbers) is a function from P into R. The usual notation for a real sequence is , where a denotes the function from P into R and a

  n

  is the value of the function at the positive integer n. The notations a1 , a2 , . . . and {a

  n

  } are also used to denote a real sequence. The number a

  n

  is called the nth term of the sequence .

  A sequence may be defined by giving an explicit formula for the nth term. For example, the formula

  defines the sequence whose value at the positive integer n is 1/n. The first three terms of this sequence are

  A sequence may also be defined inductively. Thus equations define the sequence whose first six terms are

  Consider the sequence whose nth term is defined by the formula

  The first four terms of this sequence are

  The terms corresponding to n = 100, 101, 102 are

  which are close to 1. For example, differs from 1 by only . It is clear that n/(n + 1) is “close to” 1 “for all large positive integers n.” For this reason we say that the sequence has limit 1. In general, we say that a sequence has limit L if a

  n

  is “close to” L “for all large positive integers n.” To define the limit of a sequence, we need to make the concepts “close to” and “for all large positive integers n” precise.

  Since a

  n

  − L is the distance between a

  n

  and L, we would agree that a

  n

  is “close to” L if a

  n

  − L is small. How small? Given any positive number (no matter how small) we wish to make a

  n

  − L < for all large positive integers n. If we can do this, we can make a

  n

  a

  n

  close to L as we wish simply by making small enough. “For all large positive integers n” means “for all n greater than (or equal to) some fixed positive integer N.” Thus the condition we are seeking is that if > 0, there exists a positive integer N such that a

  n

  − L < , for all n N. The positive integer N may depend on the which we are given. We would expect that as is taken smaller, we would have to choose N

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  Introduction to Statistical Limit Theory

  • Alan M. Polansky(Author)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  CHAPTER 1 Sequences of Real Numbers and Functions K. felt slightly abandoned as, probably observed by the priest, he walked by him-self between the empty pews, and the size of the cathedral seemed to be just at the limit of what a man could bear. The Trial by Franz Kafka 1.1 Introduction The purpose of this chapter is to introduce much of the mathematical limit theory used throughout the reminder of the book. For many readers, many of the topics will consist of review material, while other topics may be quite new. The study of asymptotic properties in statistics relies heavily on results and concepts from real analysis and calculus. As such, we begin with a review of limits of Sequences of real numbers and Sequences of real functions. This is followed by the development of what will be a most useful tool, Taylor’s The-orem. We then introduce the concept of asymptotic expansions, a topic that may be new to some readers and is vitally important to many of the results treated later in the book. Of particular importance is the introduction of the asymptotic order notation, which is popular in modern research in probability and statistics. The related topic of inversion of asymptotic expansions is then briefly introduced. 1.2 Sequences of Real Numbers An infinite sequence of real numbers given by { x n } ∞ n =1 is specified by the function x n : N → R . That is, for each n ∈ N , the sequence has a real value x n . The set N is usually called the index set . In this case the domain of the function is countable, and the sequence is usually thought to evolve sequen-tially through the increasing values in N . For example, the simple harmonic sequence specified by x n = n -1 has the values x 1 = 1 , x 2 = 1 2 , x 3 = 1 3 , x 4 = 1 4 , . . . , 1 2 Sequences OF REAL NUMBERS AND FUNCTIONS while the simple alternating sequence x n = ( -1) n has values x 1 = -1 , x 2 = 1 , x 3 = -1 , x 4 = 1 , .

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  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 8.3 ■ Geometric Sequences 613 ■ Infinite Geometric Series An infinite geometric series is a series of the form a  ar  ar 2  ar 3  ar 4  . . .  ar n  1  . . . We can apply the reasoning used earlier to find the sum of an infinite geometric series. The n th partial sum of such a series is given by the formula S n  a 1  r n 1  r r ? 1 It can be shown that if 0 r 0  1 , then r n gets close to 0 as n gets large (you can easily convince yourself of this using a calculator). It follows that S n gets close to a / 1 1  r 2 as n gets large, or S n S a 1  r as n S ` Thus the sum of this infinite geometric series is a / 1 1  r 2 . SUM OF AN INFINITE GEOMETRIC SERIES If 0 r 0  1 , then the infinite geometric series a ` k  1 ar k  1  a  ar  ar 2  ar 3  . . . converges and has the sum S  a 1  r If 0 r 0  1 , the series diverges. EXAMPLE 6 ■ Infinite Series Determine whether the infinite geometric series is convergent or divergent. If it is con-vergent, find its sum. (a) 2  2 5  2 25  2 125  . . . (b) 1  7 5  a 7 5 b 2  a 7 5 b 3  . . . Here is another way to arrive at the formula for the sum of an infinite geo-metric series: S  a  ar  ar 2  ar 3  . . .  a  r 1 a  ar  ar 2  . . . 2  a  rS Solve the equation S  a  rS for S to get S  rS  a 1 1  r 2 S  a S  a 1  r Fractals Many of the things we model in this book have regular predictable shapes. But recent advances in mathematics have made it possible to model such seemingly random or even chaotic shapes as those of a cloud, a flickering flame, a moun-tain, or a jagged coastline. The basic tools in this type of modeling are the fractals invented by the mathematician Benoit Mandelbrot. A fractal is a geometric shape built up from a simple basic shape by Mathematics in the Modern World scaling and repeating the shape indefinitely according to a given rule.

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