Sequences and Series

12 Key excerpts on "Sequences and Series"

• 

  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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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  10.1.1 Conceptual Recognize patterns in a sequence in order to identify the general term. The word sequence means an order in which one thing follows another in succession. In mathematics, it means the same thing. For example, if we write x, 2x 2 , 3x 3 , 4x 4 , 5x 5 , ?, what would the next term in the sequence be, the one where the question mark now stands? The answer is 6x 6 . 10.1 Sequences and Series SKILLS OBJECTIVES • Find terms of a sequence given the general term. • Apply factorial notation. • Apply recursion formulas. • Use summation (sigma) notation to represent a series and evaluate finite series and infinite series (if possible). CONCEPTUAL OBJECTIVES • Recognize patterns in a sequence in order to identify the general term. • Understand why (mn)! is not equal to m!n!. • Understand why the Fibonacci sequence is a recursion formula. • Understand why all finite series sum and why infinite series may or may not sum. In This Chapter We will first define a sequence and a series and then discuss two particular kinds of Sequences and Series called arithmetic and geometric. We will then discuss mathematical proof by induction. Finally, we will discuss the Binomial theorem, which offers us an efficient way to perform binomial expansions. • Find the general nth term of a sequence or series. • Evaluate a finite arithmetic series. • Determine if an infinite geometric series converges or diverges. • Prove a mathematical statement using induction. • Use the binomial theorem to expand a binomial raised to a positive integer power. LEARNING OBJECTIVES 10.1 Sequences and Series 899 A Sequence A sequence is a function whose domain is a set of positive integers. The function values, or terms, of the sequence are written as a 1 , a 2 , a 3 , . . . , a n , . . . Rather than using function notation, sequences are usually written with subscript (or index) notation, a subscript . A finite sequence has the domain {1, 2, 3, . . . , n} for some positive integer n.

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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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  CounterExamples

  From Elementary Calculus to the Beginnings of Analysis

  • Andrei Bourchtein, Ludmila Bourchtein(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 6 Sequences and Series 6.1 Elements of theory Numerical sequences Remark . We will consider here only infinite real sequences. Sequence definition . A sequence is a function whose domain is N . The standard notation is: a 1 , a 2 , . . . , a n , . . . or { a n } ∞ n =1 , or a n , n ∈ N or simply a n when this notation is clear within a specific context. Remark . The definition is usually extended to domains which include additional integers or exclude some naturals. The point in these extensions is to maintain the domain with the properties of ordering of N : all the elements of a sequence should be indexed (with an integer index), the first element with an initial index must exist, and each following element has the index equal to the index of the preceding element plus one. (Evidently the number of non-natural indices is finite or empty). Convergent sequence . A sequence a n is convergent if there exists a finite limit lim n →∞ a n = A . Otherwise a sequence is divergent . Remark 1 . A standard concept of the limit at infinity, slightly adopted to sequences, is used here: for every ε > 0 there exists a natural number (index) N such that for all natural numbers (indices) n > N it follows that | a n − A | < ε . Remark 2 . For the sequences the only limit point in domain is positive infinity, so the notation n → + ∞ is frequently abbreviated to n → ∞ . 175 176 Counterexamples: From Calculus to the Beginnings of Analysis Subsequences . The subsequence of a sequence a n is a new sequence composed of some elements of the original sequence, keeping their order. Monotonicity and boundedness . In the same way as for general func-tions, the properties of monotonicity (increasing/decreasing) and bounded-ness/unboundedness are defined for sequences. The usual properties (arithmetic and comparison) of the limits are valid for the convergent sequences (see the list of properties in subsection 2.1.2).

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  eBook - ePub

  An Introduction to Analysis

  • James R. Kirkwood(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  2

  Sequences of Real Numbers

  2.1 Sequences of Real Numbers

  One of the great advantages of calculus is that it enables us to solve problems of a dynamic nature, that is, problems in which a change in the variables occurs. The technique that calculus uses to deal with this type of problem is the limiting process. In this chapter we introduce sequences of real numbers. Sequences provide perhaps the simplest setting for the rigorous study of limits, and sequences will also be indispensable in studying more complex topics.

  Definition: A sequence of real numbers is a function from the positive integers into the real numbers.

  The function concept is not the most convenient way to visualize a sequence, and we shall establish a more intuitive viewpoint. If f is the function in the definition, then the range of f is the set

  { f ( 1 ) , f ( 2 ) , … } .

  The numbers

  f ( 1 ) , f ( 2 ) , …

  are called the terms of the sequence and

  f ( n )

  is called the nth term of the sequence. The domain of a sequence is always the positive integers. Therefore if we list only the range or terms of the sequence in their natural order of appearance, then the sequence will be completely described. It is customary to further simplify the notation and write f

  n

  for

  f ( n )

  . So now our sequence is written as a subscripted variable within braces

  {

  f 1

  ,

  f 2

  , … }

  or more often as

  {

  f n

  }

  . Thus

  {

  x n

  }

  will represent a sequence whose first term is x1 , whose second term is x2 , and so on.

  Sequences are often viewed as an infinite string of numbers. Two sequences are equal if and only if they are equal term by term. That is, not only must the numbers in the sequences be the same, but they must also appear in the same order. Thus the sequence

  { 1 , 2 , 3 , 4 , 5 , … }

  is not equal to the sequence

  { 2 , 1 , 3 , 4 , 5 , … }

  even though they consist of the same numbers.

  Often a sequence is represented by a function within braces, which describes the nth term of the sequence. For example, we might represent the sequence

  { 1 , 1 / 2 , 1 / 3 , 1 / 4 , … }

  as

  { 1 / n }

  , since the nth term of the sequence is equal to

  1 / n

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  Precalculus

  Functions and Graphs, Enhanced Edition

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  635 Sequences and summation notation, discussed in the first section, are very important in advanced mathematics and applications. Of special interest are arithmetic and geometric sequences, considered in Sections 9.2 and 9.3. We then discuss the method of mathematical induction, a process that is often used to prove that each statement in an infinite sequence of statements is true. As an application, we use it to prove the binomial theorem in Section 9.5. The last part of the chapter deals with counting processes that occur frequently in mathematics and everyday life. These include the concepts of permutations, combinations, and probability. Infinite Sequences and Summation Notation Arithmetic Sequences Geometric Sequences Mathematical Induction The Binomial Theorem Permutations Distinguishable Permutations and Combinations Probability 9.8 9.7 9.6 9.5 9.4 9.3 9.2 9.1 Copyright 2017 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. 636 CHAPTER 9 SEQUENCES, SERIES, AND PROBABILITY Infinite Sequences and Summation Notation 9.1 An arbitrary infinite sequence may be denoted as follows: For convenience, we often refer to infinite sequences as sequences. We may regard an infinite sequence as a collection of real numbers that is in one-to-one correspondence with the positive integers. Each number is a term of the sequence. The sequence is ordered in the sense that there is a first term , a second term , a forty-fifth term , and, if n denotes an arbitrary positive integer, an n th term . Infinite sequences are often defined by stating a for-mula for the n th term.

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  eBook - ePub

  Calculus: The Logical Extension of Arithmetic

  • Seymour B. Elk(Author)
  • 2016(Publication Date)
  • Bentham Science Publishers

    (Publisher)

  Infinite Sequences and Series

  Abstract

  Up to this point in the proposed new perspective for understanding “what is calculus?” the domain associated with both integration and differentiation has mostly been confined to continuous functions. As a concluding chapter of this opus, the focus is directed to a discussion of discrete variables with an examination of the domain of Sequences and Series; then a re-definition of important functions, in particular trigonometric and exponential functions, in term of infinite series, and a broad look at the concept of infinity as both a cardinal and an ordinal number. This chapter begins by defining the concept of sequences and both the mathematical limitations and the heuristic expectations that are fundamental to a quantitative, as well as a qualitative, development of the question “is the sequence of counting numbers unending?” and the related question “if there is such a “last” number, to which the name “infinity” has been given, what are its properties?” In the preceding chapters one observed that infinite concepts applied not only to being “infinitely large”, but also to being “infinitely small”. To this latter category the term “infinitesimal” was applied. In this chapter, the further concept, referred to as different “orders” of infinity, will be encountered. Emphasis will be placed on a concept that this author prefers to associate with the heuristic of being “infinitely dense”, in contradistinction to one of being “infinitely large”.

  7.1. INTRODUCTION TO SEQUENCES

  After a superficial introduction in Section 1.1, the attention of this treatise has been focused almost exclusively on functions in a space which have the property that between any two points there is always another point. This has led to development in that mathematical field which has the heuristic concept of “continuity”, along with the further concept of differentiability. In this chapter the initial focus will be on a domain in which the functions are discrete, rather than continuous. For such a field a different concept of infinity will be involved.

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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  815 © Andrey_Popov/Shutterstock.com Sequences and Series 10 10.1 Arithmetic Sequences 10.2 Geometric Sequences 10.3 Series Many situations make sense only for natural number inputs. In this chapter, we will investigate some of these types of models. Sequences and Series can model annual salaries, epidemics, and many other situations. 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. C H A P T E R 1 0 S e q u e n c e s a n d S e r i e s 816 Introduction to Sequences We now examine a new concept called a sequence. A sequence is an ordered list. An alphabetical class roster and a list of the number of hours worked each week of a month are both examples of sequences. To investigate the idea of a sequence further, consider the following scenario. Arithmetic Sequences LEARNING OBJECTIVES Compute and graph terms of a sequence. Identify an arithmetic sequence. Find the general term of an arithmetic sequence. Solve applied problems that involve arithmetic sequences. 10.1 Suppose you just got hired by a new company and you are earning a salary of $38,500 for the first year. Your employer tells you that you have two options to choose from: a $1000 raise each year or a 2.5% raise each year. How much will I get paid? CONCEPT INVESTIGATION Option 1 ● $1000 raise each year. To calculate the next year’s salary, add $1000. 38,500 1 1000 5 39,500 Option 2 ● 2.5% raise each year. To calculate the next year’s salary: 1. To find the amount of the raise, multiply the current salary by 2.5%. 0.025 # 38,500 5 962.50 2. Add the raise to the current salary.

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  eBook - ePub

  Introduction to Functions of a Complex Variable

  • J. H. Curtiss(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  2Sequences and Series

  2.1 Basic Definitions

  Let X be a nonempty set. A sequence in X (or from X) is a function from the set Z of real integers into X with domain of the form {k: k > n} or {k: k < n}or {k: m < k < n}, where m, n, k denote real integers. In the last case, the sequence is called a finite sequence. The ordering in Z automatically induces an ordering on the values of the sequence. Thus, a finite sequence with, say, N values is an ordered N-tuple.

  In specifying a sequence by its values, it is customary to use a subscript notation such as

         

  〈

  x k

  〉

  k = n

  ∞

           

  . If the intended domain is N = {k: k = 0, 1, 2, . . .}, we abbreviate the notation to ⟨xk ⟩. This notation is used in the case of other domains when no confusion results. When the pattern is clear, a sequence is often informally specified by writing down the first few terms; thus, x0 , x1 , x2 ,.... We indicate the range of the sequence ⟨xk ⟩ by {xk }, with appropriate elaboration if the exact domain of the sequence must be emphasized. The range of a sequence is sometimes called the trace.

  A subsequence of

      〈

  x k

  〉 n ∞

       

  is a sequence of the form ⟨xϕ ( k ) ⟩, where ϕ is a function from [n, ∞) into [n, ∞) satisfying ϕ(h) < ϕ(k), h < k.

  If X is a field, it is consistent with the definitions in Sec. 1.7 to define the sum of two sequences ⟨xk ⟩, ⟨yk ⟩ written ⟨xk ⟩ + ⟨yk ⟩, by ⟨xk + yk ⟩. The product ⟨xk ⟩ · ⟨yk ⟩ is defined by ⟨xk · yk ⟩ and if yk ≠ 0, k = 0, 1, 2, . . ., the quotient ⟨xk ⟩/⟨yk ⟩ is defined by ⟨xk /yk ⟩.

  The spaces X in which the sequences in this book lie are C, R, and certain spaces of complex-valued functions.

  2.2 Metric Spaces

  The concept of convergence and divergence is central in the theory of sequences. It can be treated at various levels of abstraction, but since we specialize in sequences in the spaces mentioned previously, an appropriate general setting for the basic definitions is given by the concept of a metric space.

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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)

  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. Similarly,  3 = 470,  4 = 460, and  5 = 450. Arithmetic Sequences You buy a used car that has already been driven 15,000 miles and drive it 8000 miles per year. The odometer registers 23,000 miles 1 year after your purchase, 31,000 miles after 2 years, and so on. The yearly odometer readings form a sequence   whose terms are 15,000, 23,000, 31,000, 39,000, … . 1 www.statista.com/statistics/201182/forecast-of-smartphone-users-in-the-us/, accessed March 15, 2017. 13.1 SEQUENCES 13-3 Each term of the sequence is obtained from the previous term by adding 8000; that is, the difference between successive terms is 8000. A sequence in which the difference between pairs of successive terms is a fixed quantity is called an arithmetic sequence. Example 4 Which of the following sequences are arithmetic? (a) 9, 5, 1, −3, −7 (b) 3, 6, 12, 24, 48 (c) 2, 2 + , 2 + 2, 2 + 3 (d) 10, 5, 0, 5, 10 Solution (a) Each term is obtained from the previous term by subtracting 4. This sequence is arithmetic. (b) This sequence is not arithmetic: each term is twice the previous term. The differences are 3, 6, 12, 24. (c) This sequence is arithmetic:  is added to each term to obtain the next term. (d) This is not arithmetic. The difference between the second and first terms is −5, but the difference between the fifth and fourth terms is 5. We can write a formula for the general term of an arithmetic sequence. Look at the sequence 2, 6, 10, 14, 18, …, in which the terms increase by 4, and observe that  1 = 2  2 = 6 = 2 + 1 ⋅ 4  3 = 10 = 2 + 2 ⋅ 4  4 = 14 = 2 + 3 ⋅ 4.

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  6.2 A RITHMETIC S EQUENCES The Definition of an Arithmetic Sequence We’ve already mentioned that a sequence is an ordered progression of real num-bers that may or may not follow a prescribed pattern. If the order is changed, the sequence is changed, so the order of those numbers is fixed once it has been established. Let’s consider a particular type of sequence whose terms follow a specific sort of pattern. The pattern will probably be evident, but we’ll point it out to ensure clarity. { a n } 5 {4, 9, 14, 19, . . .} The terms of the sequence happen to satisfy the condition that they increase by a particular constant value (in this case, 5) from term to term. Such a sequence is called an arithmetic sequence . The emphasis in the word “arithmetic” is on the “met” fragment rather than the “rith” you may be used to using, so the pronunciation is “arith met ic sequence.” Copyright 2013 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. Sequences and Series 223 The constant value that is the difference between the terms of an arithmetic sequence is, appropriately, referred to as the common difference for the terms of the sequence, and we say that the terms follow an arithmetic progression . A useful fact about such sequences is that they are completely determined by the value of the first term, a 1 , and the common difference, which we can refer to with the variable d . Let’s examine the terms of an arithmetic sequence a bit more closely by con-sidering the example already presented.

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  Algebra

  A Combined Course 2E

  • Charles P. McKeague(Author)
  • 2018(Publication Date)
  • XYZ Textbooks

    (Publisher)

  There is an interesting relationship between the sequence of odd numbers and the sequence of squares that is found by adding the terms in the sequence of odd numbers. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 When we add the terms of a sequence the result is called a series. DEFINITION series The sum of a number of terms in a sequence is called a series. A sequence can be finite or infinite, depending on whether the sequence ends at the nth term. For example, 1, 3, 5, 7, 9 is a finite sequence, but 1, 3, 5, . . . is an infinite sequence. Associated with each of the preceding sequences is a series found by adding the terms of the sequence: 1 + 3 + 5 + 7 + 9 Finite series 1 + 3 + 5 + . . . Infinite series A Summation Notation for a Series In this section, we will consider only finite series. We can introduce a new kind of notation here that is a compact way of indicating a finite series. The notation is called summation notation, or sigma notation because it is written using the Greek letter sigma. ©iStockphoto.com/dzphotovideo Chapter 15 Sequences and Series 1160 The expression ∑ i = 1 4 (8i − 10) is an example of an expression that uses summation notation. The summation notation in this expression is used to indicate the sum of all the expressions 8i − 10 from i = 1 up to and including i = 4. That is, ∑ i = 1 4 (8i − 10) = (8 ⋅ 1 − 10) + (8 ⋅ 2 − 10) + (8 ⋅ 3 − 10) + (8 ⋅ 4 − 10) = −2 + 6 + 14 + 22 = 40 The letter i as used here is called the index of summation, or just index for short. Here are some examples illustrating the use of summation notation. EXAMPLE 1 Expand and simplify ∑ i = 1 5 (i 2 − 1). Solution We replace i in the expression i 2 − 1 with all consecutive integers from 1 up to 5, including 1 and 5. ∑ i = 1 5 (i 2 − 1) = (1 2 − 1) + (2 2 − 1) + (3 2 − 1) + (4 2 − 1) + (5 2 − 1) = 0 + 3 + 8 + 15 + 24 = 50 EXAMPLE 2 Expand and simplify ∑ i = 3 6 (−2) i .

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