Mathematics
Series Maths
In mathematics, a series is the sum of the terms of a sequence. It is often represented using sigma notation and can be finite or infinite. The study of series involves understanding convergence, divergence, and methods for finding the sum of a series, such as using formulas or techniques like telescoping.
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5 Key excerpts on "Series Maths"
eBook - PDFCalculus
Concepts and Contexts, Enhanced Edition
- James Stewart(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
553 Infinite Sequences and Series Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zeno’s paradoxes and the decimal representation of numbers. Their importance in calculus stems from Newton’s idea of representing functions as sums of infinite series. For instance, in finding areas he often integrated a function by first expressing it as a series and then integrating each term of the series. We will pursue his idea in Section 8.7 in order to integrate such functions as . (Recall that we have previously been unable to do this.) Many of the functions that arise in mathematical physics and chemistry, such as Bessel func-tions, are defined as sums of series, so it is important to be familiar with the basic concepts of convergence of infinite sequences and series. Physicists also use series in another way, as we will see in Section 8.8. In studying fields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represents it. e x 2 8 Goran Bogicevic/Shutterstock.com 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. 554 CHAPTER 8 INFINITE SEQUENCES AND SERIES A sequence can be thought of as a list of numbers written in a definite order: The number is called the first term, is the second term, and in general is the nth term. We will deal exclusively with infinite sequences and so each term will have a successor .
- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 724 CHAPTER 11 Sequences, Series, and Power Series Sequences Many concepts in calculus involve lists of numbers that result from applying a process in stages. For example, if we use Newton’s method (Section 4.8) to approximate the zero of a function, we generate a list or sequence of numbers. If we compute average rates of change of a function over smaller and smaller intervals in order to approximate an instan- taneous rate of change (as in Section 2.7), we also generate a sequence of numbers. In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concern- ing space and time that were held in his day. In one of his paradoxes, Zeno argued that a man standing in a room could never walk to a wall because he would first have to walk half the distance to the wall, then half the remaining distance, and then again half of what still remains, continuing in this way indefinitely (see Figure 1). The distances that the man walks at each stage form a sequence: 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , . . . , 1 2 n , . . . 1 2 1 4 1 2 n . . . . . . ■ Infinite Sequences An infinite sequence, or just a sequence, can be thought of as a list of numbers written in a definite order: a 1 , a 2 , a 3 , a 4 , . . . , a n , . . . The number a 1 is called the first term, a 2 is the second term, and in general a n is the n th term. We will deal exclusively with infinite sequences and so each term a n will have a successor a n11 .
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Attempting to remember the complicated form without understanding its structure is much more difficult and is not necessary, even when it comes to solving a typical rather formal problem. For example, suppose that we are asked to find the last term in an arithmetic series whose sum is 222, with a first term of 2 and a common difference of 3. To do this we need to find and solve an equation to give the number of terms in the series which can be done as shown below: The last, or nth, term Mean term Sum of first n terms Equation for sum: Solving the equation: The only sensible solution is n = 12, since the number of terms must be a positive whole number, and so, by substituting into 3n - 1, the last term is 35. Arithmetic sequences and series are essentially very simple to understand and use if the necessary formulae are presented in a simple, possibly verbal, form which can be related easily to numerical examples. The procedure above has been designed to make each step towards the solution meaningful in itself, and easy to check. A blind substitution of numbers into a complicated looking formula may lead straight to the equation of the fourth line, but greater insight is obtained by leading up to it in the way shown. When students are confident with ideas they can modify and abbreviate their approach to problems, but for many it is better to use a slightly lengthier argument whose components are understood than a shorter one that relies just on memorized results. GEOMETRIC SEQUENCES AND SERIES Starting a new topic with a problem and letting students arrive at key results for themselves is potentially more motivating and can make them more memorable. The pocket money problem is one way of introducing some of the ideas of geometric sequences and series. The plan is for students to propose to their parents a plausible sounding pocket money scheme that turns out to be very rewarding.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
6 Series and limits Many examples exist in the physical sciences of situations where we are presented with a sum of terms to evaluate. As just two examples, there may be the need to add together the contributions from successive slits in a diffraction grating in order to find the total light intensity at a particular point, or to compute, for a particular site in a crystal, the electrostatic potential due to all the other ions in the crystal. 6.1 Series • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • A general series may have either a finite or infinite number of terms. In either case, the sum of the first N terms of a series (often called a partial sum) is written S N = u 1 + u 2 + u 3 + · · · + u N , where the terms of the series u n , n = 1 , 2 , 3 , . . . , N , are numbers that may in general be complex. If some or all of the terms are complex then, in general, S N will also be complex, and we can write S N = X N + iY N , where X N and Y N are the partial sums of the real and imaginary parts of each term separately and are therefore real. If a series has only N terms then the partial sum S N is of course the sum of the series. Sometimes we encounter series where each term depends on some variable, x , say. In this case the partial sum of the series will depend on the value assumed by x . For example, consider the infinite series S ( x ) = 1 + x + x 2 2! + x 3 3! + · · · This is an example of a power series; these are discussed in more detail in Section 6.5 . It is in fact the Maclaurin expansion of exp x (see Section 6.6.3 and Appendix A ). Therefore, S ( x ) = exp x and, of course, its value varies according to the value of the variable x . A series might just as easily depend on a complex variable z . A general, random sequence of numbers can be described as a series and a sum of the terms found.
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 762 CHAPTER 11 Sequences, Series, and Power Series Sequences Many concepts in calculus involve lists of numbers that result from applying a process in stages. For example, if we use Newton’s method (Section 3.8) to approximate the zero of a function, we generate a list or sequence of numbers. If we compute average rates of change of a function over smaller and smaller intervals in order to approximate an instan- taneous rate of change (as in Section 2.1), we also generate a sequence of numbers. In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concern- ing space and time that were held in his day. In one of his paradoxes, Zeno argued that a man standing in a room could never walk to a wall because he would first have to walk half the distance to the wall, then half the remaining distance, and then again half of what still remains, continuing in this way indefinitely (see Figure 1). The distances that the man walks at each stage form a sequence: 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , . . . , 1 2 n , . . . 1 2 1 4 1 2 n . . . . . . ■ Infinite Sequences An infinite sequence, or just a sequence, can be thought of as a list of numbers written in a definite order: a 1 , a 2 , a 3 , a 4 , . . . , a n , . . . The number a 1 is called the first term, a 2 is the second term, and in general a n is the n th term. We will deal exclusively with infinite sequences and so each term a n will have a successor a n11 .
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