Binomial Series

3 Key excerpts on "Binomial Series"

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  eBook - PDF

  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  586 20 Sequences, Series, and the Binomial Theorem ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to • Identify various types of sequences and series. • Write the general term or a recursion relation for many series. • Compute any term or the sum of any number of terms of an arithmetic progression or a geometric progression. • Compute any term of a harmonic progression. • Insert any number of arithmetic means, harmonic means, or geometric means between two given numbers. • Compute the sum of an infinite geometric progression. • Compute, graph, and find sums of sequences by calculator. • Solve applications problems using series. • Raise a binomial to a power using the binomial theorem. • Find any term in a binomial expansion. ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ We start this chapter with a general introduction to sequences and series. Sequences and series are of great interest because a computer or calculator uses series internally to calculate many functions, such as the sine of an angle, the logarithm of a number, and so on. We then cover some specific sequences, such as the arithmetic progres- sion and the geometric progression. Progressions are used to describe such things as the sequence of heights reached by a swinging pendulum on subsequent swings or the monthly balance on a savings account that is growing with compound interest. We finish this chapter with an introduction to the binomial theorem. The bino- mial theorem enables us to expand a binomial, such as without actu- ally having to multiply the terms. It is also useful in deriving formulas in calculus, statistics, and probability. (3x 2  2y) 5 , Section 1 ◆ Sequences and Series 587 20–1 Sequences and Series Before using sequences and series, we must define some terms. Sequences A sequence is a set of quantities, called terms, which follow each other in a definite order.

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  eBook - PDF

  Maths: A Student's Survival Guide

  A Self-Help Workbook for Science and Engineering Students

  • Jenny Olive(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  7 Binomial Series and proof by induction In this chapter we find out how to do binomial expansions, and see how they can describe some real-life situations. We also look at a new method of proving mathematical statements. The chapter is divided into the following sections. 7.A Binomial Series for positive whole numbers (a) Looking for the patterns, (b) Permutations or arrangements, (c) Combinations or selections, (d) How selections give binomial expansions, (e) Writing down rules for binomial expansions, (f) Linking Pascal’s Triangle to selections, (g) Some more binomial examples 7.B Some applications of Binomial Series and selections (a) Tossing coins and throwing dice, (b) What do the probabilities we have found mean? (c) When is a game fair? (Or are you fair game?) (d) Lotteries: winning the jackpot . . . or not 7.C Binomial expansions when n is not a positive whole number (a) Can we expand (1 + x ) n if n is negative or a fraction? If so, when? (b) Working out some expansions, (c) Dealing with slightly different situations 7.D Mathematical induction (a) Truth from patterns – or false mirages? (b) Proving the Binomial Theorem by induction, (c) Two non-series applications of induction 7.A Binomial Series for positive whole numbers 7.A. (a) Looking for the patterns The first half of this chapter describes what are called Binomial Series. I have given them so much space because they have many applications. For this reason it is important that you should be able to do binomial expansions correctly and happily. The word ‘binomial’ comes from the two quantities put together in a bracket which we start from. Binomial expansions are what we get when we raise these brackets to different powers and then multiply the brackets together to find the result. In this first section all these powers will be positive whole numbers. Here are some examples. ( a + b ) 1 is just a + b ( a + b ) 2 = ( a + b )( a + b ) = a 2 + 2 ab + b 2 .

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  Mathematics for Engineering

  • W Bolton, W. Bolton(Authors)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  Section CFurther algebra

  7 Sequences and series 8 Exponentials 9 Logarithms The aims of this section are to enable the reader to:

  • Use sequences and the arithmetic, geometric and Binomial Series in the solution of engineering problems.
  • Use the binomial theorem to simplify algebraic expressions.
  • Manipulate and solve equations involving logarithmic and exponential expressions.
  • Solve engineering problems involving exponential growth and decay.

  This section extends the coverage of algebra given in Section B . It is assumed that Section B has been covered and the parts of Chapter 1 concerned with powers fully comprehended. There is a case for studying parts of Section E on graphs prior to Section C and others parts immediately following it.

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  7 Sequences and series

  7.1 Introduction

  This chapter is an introduction to sequences and series. A sequence is a set of numbers arranged in a definite order. Consider the following sequence of numbers: 1, 3, 5, 7, 9. Such a set of numbers is termed a sequence because they are stated in a definite order, i.e. 1 followed by 3 followed by 5, etc. In the above sequence of 1, 3, 5, 7, 9, each successive member is formed by adding 2 to the previous member. Another example of a sequence is money in an interest-earning bank account. Initially there might be, say, £100 in the account. If it earns interest at the rate of, say, 10% per year then after 1 year there will be £110 in the account. At the end of the next year there will be £121 in the account. At the end of the next year £132.1. The sequence of numbers describing the amount in the account at the ends of each year are:

  £100 £110 £121 £132.1 We have a sequence of numbers where there is a definite relationship between successive values.

  The term series is used for the sum of the terms of a sequence. There are many instances in mathematics were we have values which we can express as series and so aid calculations. A particular useful series is termed the Binomial Series

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