Limit of a Sequence

11 Key excerpts on "Limit of a Sequence"

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  eBook - PDF

  Fundamentals of Abstract Analysis

  • Andrew Gleason(Author)
  • 2018(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  CHAPTER 12 LIMITS Analysis is distinguished as a branch of mathematics by its frequent appeals to the notion of limit. Differentiation, integration, and infinite summation are all applications of the limit concept. So far we have used this concept only twice, in showing the existence of square roots of positive numbers and in proving that any two complete ordered fields are isomorphic. Both of these proofs appealed directly to the completeness of the real numbers. We shall now develop a theory of limits for both real and complex numbers which will serve as a prototype for a more general theory in Chapter 14. 12-1. CONVERGENT SEQUENCES 12-1.1. Definition. Let X be any set. A sequence of members of X is a function from N to X. Elements in the range of a sequence are usually called terms of the sequence. Sometimes they are called members or elements of the sequence. A sequence is a special kind of family. We shall apply the term sequence to a function with domain N only when the emphasis is strongly on the values taken by the function and the order in which they appear. The intuitive picture is of infinitely many elements written down one after another, possibly with repetitions. This idea is carried into the notation, for one often writes something like, “Let t/i, u 2, u 3, . . . be a sequence of . . . , ” which is really a substitute for the formal “Let u be a function (argument written as a subscript) from N to . . .” We shall follow the common practice of writing a dummy argument for sequences. Thus we shall write {xn} for the sequence informally denoted by jci , x 2, * 3, . . . This is an abbreviation of the general notation {xn n e N} described in 3-4.18. The term sequence is also applied more generally to denote a function whose domain is any set of consecutive integers. There are, therefore, finite sequences, which have finite domains, and sequences with all of I as domain.

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  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Limit of a Sequence (less precise version) A sequence has limit L if from some point on, all the terms of the sequence are very close to L. This definition fails to be precise because the phrase “very close” is too vague. A more precise definition of limit will be given soon, but first we examine some examples to get a feel for what is meant by taking the Limit of a Sequence. Example 1 What is the limit of the sequence whose n th term equals √ n 2 + n - n? solution The Limit of a Sequence depends on the behavior of the n th term for large n √ n 2 + n - n 1 0.4142136 10 0.4880885 100 0.4987562 1000 0.4998751 10000 0.4999875 100000 0.4999988 1000000 0.4999999 values of n. The table here shows the values of the n th term of this sequence for some large values of n, calculated by a computer and rounded off to seven digits after the decimal point. This table leads us to suspect that this sequence has limit 1 2 . This suspicion is correct, as can be shown by rewriting the n th term of this sequence as follows (see Problem 29 for a hint on how to do this): This formula shows that every term in this sequence is less than 1 2 , as we should suspect from the table above. p n 2 + n - n = 1 q 1 + 1 n + 1 . If n is very large, then 1 + 1 n is very close to 1, and thus the right side of the equation above is very close to 1 2 . Hence the limit of the sequence in question is indeed equal to 1 2 . Not every sequence has a limit, as shown by the following example of the sequence of alternating 1’s and -1’s. Section 6.3 Limits 471 Example 2 Consider the sequence whose n th term is equal to (-1) n-1 . Explain why this sequence does not have a limit. solution The sequence in question is the sequence of alternating 1’s and -1’s: 1, -1, 1, -1, . . . . If this sequence had a limit, then that limit would have to be very close to 1 and very close to -1. However, no number is close to both 1 and -1. Thus this sequence does not have a limit.

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  The How and Why of One Variable Calculus

  • Amol Sasane(Author)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  (Write down the first few terms of this sequence.) Example 2.1. Here are a couple of examples of sequences, and we have also displayed the first few terms. 1 1 1 1 1 − 1 2 2 2 3 4 4 5 6 6 7 (1) n ∈ 1 , 1 , 1 , · · · 1 n n ∈ 1 , 1 2 , 1 3 , · · · (( − 1) n ) n ∈ − 1 , 1 , − 1 , − 1 , 1 , − 1 , · · · ( n ) n ∈ 1 , 2 , 3 , · · · 1 + 1 2 + 1 3 + · · · + 1 n n ∈ 1 , 1 + 1 2 , 1 + 1 2 + 1 3 , · · · ♦ 46 THE HOW AND WHY OF ONE VARIABLE CALCULUS What do we want to know about sequences? In Calculus, we want to know ‘the limiting behavior’ of the sequence, that is, what a n behaves like for large n , and in particular, whether a n gets closer and closer to some number L (called the limit of the sequence at hand). What is the motivation for studying the limiting behavior of sequences? For example, the terms of the sequence might be the sum of the areas of the rectangles in the picture on the left below, or they might be the slopes of the chords in the picture on the right, and we might be interested in the limiting behavior because we want to calculate the area under the graph (left picture) or the instantaneous rate of change of function at the point c (right picture). Thus we want to know what happens when n increases to the sequence ( a n ) n ∈ N where (Left picture) a n = n − 1 k = 1 m k · k n , here m k : = height of k th shaded rectangle, (Right picture) a n = f ( c + 1 n ) − f ( c ) 1 n . 0 1 n 2 n · · · n − 1 n 1 c c + 1 c + 1 2 2.1 Limit of a convergent sequence We want to give a precise definition for ‘the sequence ( a n ) n ∈ N is convergent with limit L ’ or ‘ lim n →∞ a n = L ’. Intuitively, by the above, we mean that there is a number L such that the terms of the sequence are getting ‘closer and closer’ or are ‘settling down’ to L for larger and larger values of n . If there is no such finite number L to which the terms of the sequence get arbitrarily close, then the sequence is said to diverge.

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  The Fundamentals of Mathematical Analysis

  • G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  If we say that the variable ranges over some sequence of values, then the reader might imagine that the variable takes its value in consecutive instances of time; in fact, however, it has nothing to do with time. Only for clarity sometimes the fol-lowing expressions are used: remote values of the variable, starting from some place or from some instant of variation, etc. 28. Definition of the Limit of a Sequence. The ordering of the values of the variable x n , according to increasing numbers which led us to consider the sequence (2) of these values, simplifies the concept of the process of the variable x n approaching its limit a— as n increases to infinity. The number a is called the limit of the function x n if the latter differs from a by an arbitrarily small amount, beginning from a certain place, i.e. for all sufficiently large numbers n. This statement clearly expresses the essence of the matter, but what arbitrarily small or sufficiently great means has to be explained. We now present a longer but comprehensive and precise definition of limit. The number a is said to be the limit of the variable x n if for any positive number , no matter how small it is, a number N exists such that all values of x n the numbers of which n > N satisfy the inequality χ η -α<ε. (3) The fact that a is the limit of variable x n is written as follows: lim x n = a (lim is an abbreviation of the Latin word limes, meaning limit). Sometimes it is said that the variable tends to a and we then write x n -+ a. § 1. LIMIT OF A FUNCTION 55 Finally, the number a is also called the limit of sequence (2) and we may say that this sequence converges to a. The inequality (3) where ε is arbitrary is the precise statement of the fact that x n differs from a by an arbitrarily small amount, and the number N indicates the place beginning from which this fact occurs, so that all numbers n >N are sufficiently large.

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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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  Biocalculus

  Calculus, Probability, and Statistics for the Life Sciences

  • James Stewart, Troy Day, James Stewart(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  . . and for larger values of n we have a 10 -0.1, a 100 -0.01, a 1000 -0.001, a 1,000,000 -0.000001, . . . The larger the value of n , the smaller the value of a n . The terms are approaching 0 as n increases. [See Figure 1(a).] (b) Here the terms are 2 1, 1, 2 1, 1, 2 1, 1, 2 1, . . . The values of the terms alternate between 1 and 2 1 forever. So they don’t approach any fixed number. [See Figure 1(b)]. ■ The sequences in Example 1 behave quite differently. The terms a n -1 y n approach 0 as n becomes large. (In fact we could make 1 y n as small as we like by taking n large enough). We indicate this by saying that the sequence has limit 0 and by writing lim n l ` 1 n -0 On the other hand, the sequence b n -s 2 1 d n does not have a limit, that is, lim n l ` s 2 1 d n does not exist ■ Definition of a Limit In general we write lim n l ` a n -L if the terms a n approach L as n becomes large. 1 _1 1 n b n (b) b =(_1) n n n 1 0 1 a n (a) a = n n 1 FIGURE 1 T HE IDEA OF A LIMIT is the basic concept in all of calculus. It underlies such phenomena as the long-term behavior of a population, the rate of growth of a tumor, and the area of a leaf. Copyright 2016 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. SECTION 2.1 | Limits of Sequences 91 (1) Definition A sequence h a n j has the limit L and we write lim n l ` a n -L or a n l L as n l ` if we can make the terms a n as close to L as we like by taking n sufficiently large. If lim n l ` a n exists, we say the sequence converges (or is convergent ). Otherwise, we say the sequence diverges (or is divergent ).

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  Real Analysis

  A Constructive Approach Through Interval Arithmetic

  • Mark Bridger(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  3. LIMITS, SEQUENCES AND SERIES 3.1 Sequences and Convergence In section 1.8 we talked about limiting families and their limits, which are real numbers. In this section we talk about sequences of reals and their limits. This provides a parallel theory which, in a sense to be made precise, is equivalent to that of limiting families. Intuitively, a sequence is just a list of real numbers; for example, 1 > 2 > 4 > 8 > = = = 1 > 1 @ 2 > 1 @ 4 > 1 @ 8 > = = = 2 > s 2 > q s 2 > r q s 2 > = = = are all sequences. However, we want to be able to describe the various terms of the sequence precisely. For example, in the last example above, the terms represent a certain number of iterated square-roots of 2 . Thus, given some whole number n , we want a notation for the n th term of the sequence. That is, we want to be able to make an association n $ n th term of the sequence. This suggests that a sequence should be a function which associates with each natural number n some real number: the n th term of the sequence. Recall that the the set of non-negative whole numbers is denoted by N ( N = { 0 > 1 > 2 > 3 > === } ). Definition 3.1.1 A sequence d of real numbers is a function d : N $ R . Notation 3.1.2 It is customary to denote d ( n ) by d n and the sequence itself by ( d n ) . Also, sometimes d 0 is omitted, so that the rst term of the sequence is d 1 . Definition 3.1.3 A sequence of real numbers ( d n ) is said to converge to the real number O if, for any A 0 there is an integer Q d ( ) 0 such that | d l O | whenever l Q d ( ) . O is called the limit of the sequence, and we write ( d n ) $ O , or O = lim n $4 ( d n ) . Q d ( ) is called a modulus of convergence for ( d n ) . Remark 3.1.4 In the above de nition we have used weak inequality: | d n O | whenever n Q d ( ) . We might have phrased the de nition using strong inequality, that is, | d n O | ? whenever n Q d ( ) . This would lead to the notion of a strong 99 100 LIMITS, SEQUENCES AND SERIES modulus of convergence.

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  Introduction to Statistical Limit Theory

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

    (Publisher)

  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 , . . . . One can also consider real sequences of the form x t with an uncountable domain such as the real line which is specified by a function x t : R → R . Such sequences are essentially just real functions. This section will consider only real sequences whose index set is N . The asymptotic behavior of such sequences is often of interest. That is, what general conclusions can be made about the behavior of the sequence as n becomes very large? In particular, do the values in the sequence appear to “settle down” and become arbitrarily close to a single number x ∈ R as n → ∞ ? For example, the sequence specified by x n = n -1 appears to become closer and closer to 0 as n becomes very large. If a sequence has this type of property, then the sequence is said to converge to x as n → ∞ , or that the limit of x n as n → ∞ is x , usually written as lim n →∞ x n = x, or as x n → x as n → ∞ . To decide whether a given sequence has this type of behavior, a mathematical definition of the convergence or limiting concept is required. The most common definition is given below. Definition 1.1. Let { x n } ∞ n =1 be a sequence of real numbers and let x ∈ R be a real number. Then x n converges to x as n → ∞ , written as x n → x as n → ∞ , or as lim n →∞ x n = x, if and only if for every ε > 0 there exists n ε ∈ N such that | x n -x | < ε for every n ≥ n ε . This definition ensures the behavior described above. Specify any distance ε > 0 to x , and all of the terms in a convergent sequence will eventually be closer than that distance to x . Example 1.1. Consider the harmonic sequence defined by x n = n -1 for all n ∈ N . This sequence appears to monotonically become closer to zero as n increases.

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  Available until 14 Apr |Learn more

  Advanced Calculus

  Theory and Practice

  • John Petrovic(Author)
  • 2013(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The convergence (or divergence) of such a sequence means, by definition, that the series converges (or diverges). Therefore the series ∑ ∞ n =1 sin n 2 n and ∑ ∞ n =1 1 n 2 are convergent, while the series ∑ ∞ n =1 1 n diverges. As we will see later, one of the major roles of sequences will be in the study of infinite series. Sequences and Their Limits 25 Did you know? Theorem 1.6.6 was proved by Cauchy in Cours d’analyse in 1821. Four years earlier, Bernard Bolzano (1781–1848), a Bohemian mathematician, philosopher, and a Catholic priest, explicitly stated the same result and gave an incomplete proof. It is very likely that Cauchy was aware of this work. Nevertheless, it should be emphasized that Cours d’analyse remains one of the most important and influential mathematics books ever written. Much of the precision and rigor that is nowadays an essential part of mathematics dates to this book. Problems In Problems 1.6.1–1.6.5, use the definition to determine which of the following sequences is a Cauchy sequence: 1.6.1 . a n = arctan 1 2 + arctan 2 2 2 + · · · + arctan n 2 n . 1.6.2 . a n = 1 + 1 4 + 2 2 4 2 · · · + n 2 4 n . 1.6.3 . a n = 1 2 2 + 2 3 2 + · · · + n ( n + 1) 2 . 1.6.4 . a n = √ n . 1.6.5 . a n = n + 1 n . 1.6.6 . Let { a n } be a sequence such that | a n +1 − a n | → 0. Prove or disprove: { a n } is a Cauchy sequence. 1.6.7 . Let 0 0, and suppose that { a n } is a sequence such that, for all n ∈ N , | a n +1 − a n | ≤ Mr n . Prove that { a n } is a Cauchy sequence. 1.6.8 . Suppose that { a n } is a sequence such that, for all n ∈ N , | a n | < 2 , and | a n +2 − a n +1 | ≤ 1 8 | a 2 n +1 − a 2 n | . Prove that { a n } is a Cauchy sequence. 1.7 Limit Superior and Limit Inferior It is often the case that we are dealing with a divergent sequence, yet we would like to be able to make a statement about its long-term behavior. For example, it was established in Example 1.3.6 that the sequence a n = ( − 1) n is divergent.

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  The Real Analysis Lifesaver

  All the Tools You Need to Understand Proofs

  • Raffi Grinberg(Author)
  • 2017(Publication Date)
  • Princeton University Press

    (Publisher)

  Sequences C H A P T E R 14 Convergence We’ll begin our study of sequences and series by exploring the notion of convergence , which basically asks, “as a sequence continues to infinity, does it get arbitrarily close to some point?” Though we introduced sequences in Chapter 2, we should first give a more formal definition, which is based on what we learned about functions in Definition 8.3. Definition 14.1. (Sequence) A sequence in a metric space X is a function f : N → X , f : n → s n where p n ∈ X , ∀ n ∈ N . We denote the sequence as { p n } or as p 1 , p 2 , . . . The set of all possible values of the sequence is called the range of { p n } . The sequence is bounded if its range is bounded in X (according to Definition 9.3). Example 14.2. (Sequences) By this definition, every countable set can be made into a sequence. In particular, Q can be arranged into a sequence, but R cannot. Remember that all sequences go on for an infinite number of steps, but the range of a sequence might be finite if the sequence repeats. For example, the set { 1 } is not a sequence in and of itself, but we can make it into a sequence by writing 1 , 1 , 1 , . . . Let’s look at the following sequences in the metric space R : 1. If s n = 1 n for every n ∈ N , then the range of { s n } is the set { 1 , 1 2 , 1 3 , . . . } . This range is infinite and bounded (since the distance between any two elements is ≤ 1). Thus we say that { s n } is bounded. 2. If s n = n 2 for every n ∈ N , then the range of { s n } is the set { 1 , 4 , 9 , . . . } . This range is infinite and unbounded (since the numbers get larger and larger). Thus we say that { s n } is unbounded. 3. If s n = 1 + ( − 1) n n for every n ∈ N , then the range of { s n } is the set { 0 , 2 3 , 4 5 , 6 7 , . . . } ∪ { 3 2 , 5 4 , 7 6 , . . . } . This range is infinite and bounded (since the distance between any two elements is ≤ 3 2 ). Thus we say that { s n } is bounded.

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  Introduction to Mathematics with Maple

  • P Adams, K Smith;R V??born??;;(Authors)
  • 2004(Publication Date)
  • WSPC

    (Publisher)

  Obviously this might be difficult or not feasible. It is, however, possible to prove powerful theorems n-00 Limits of Sequences 243 concerning limits from the definition, and then to use these theorems to find limits. Some such theorems are given in the next section. (F) Consider now a real sequence x, that is, a sequence whose terms are real numbers Xn. Since the inequality Ixn - 11 < E is equivalent by Theorem 4.4 to all terms of the sequence with an index greater than N lie in the interval 31 - E , 1 + E [ . This means that outside this interval can lie only terms of the sequence with n 5 N , which are finitely many. If the terms xn are graphed as points on the number line then they cluster near 1 in the interval 11 - E , 1 + E [ (see Figure 10.1). Fig. 10.1 Limit of a real sequence For a complex sequence x, that is, a sequence xn E C for all n, the inequality lxn - 11 < E means that the distance between xn and I is less than E. Geometrically this means that, in the complex plane, xn lies in the circle centered at I and of radius E. If the sequence has a limit 1 the points xn in the complex plane cluster near 1 in the circle centered at 1 and of radius E , see Figure 10.2, where the terms of the sequence starting with 2 5 lie within the circle. It goes without saying that the letter n in the symbol lim Xn = I can be replaced by any other letter without altering the meaning. Thus lim Xn = lim xk = lim Xm. Sometimes We shall not do that in this book. If several letters are involved in the formula for xn, confusion can arise. It is not clear what is the mean- ing of lim ( n - + i). It is left as an exercise to show that n-+m n+m k+m m-mo lim xn is abbreviated to limx,. n+m lim (5 + i) = 1 + 2 n n-+m n + 2 (J> u The notation xn ' -+ 1 has the same disadvantage as abbreviatinc 244 Introduction to Mathematics with Maple Complex plane 0 Fig.

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