{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

L7Sequences - Introduction to Microeconomics Lecture 7...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Microeconomics Lecture 7 Maths: Sequences, Series and Limits; the Economics of Savings & Investment Ian Crawford
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Outline Sequences and series Arithmetic and geometric Deposits and interest Present Value Savings and borrowing decisions Limits The number e and some applications
Image of page 2
Sequences: order set of numbers A SEQUENCE is a set of numbers arranged in a definite order, e.g. (i) 2, 4, 6, 8,…; (ii) 1, 4, 9, 16, 25,… The dots indicate that these are infinite sequences. Sequences may be finite. There may be a formula that defines the terms e.g. for sequence (ii) the n th term, u n , is u n = n 2 A formula may use the preceding terms: “a recurrence relation”. An example is u 1 = 1, u 2 = 1, u n = u n- 1 + u n- 2 This gives the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,…
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Series: sum of a sequence A SERIES is formed when the terms of a sequence are added together. Notation: Here the notation Σ (pronounced “sigma”) denotes “the sum of” 1 2 1 ... n r n r u u u u = + + +
Image of page 4
Series: sum of a sequence Some times it’s handy to express a product as a sum. Notation: Here the notation ∏ denotes “the product of” By taking logs it becomes a sum: 1 2 1 ... = × × × n r n r u u u u 1 1 ln ln = = n n r r r r u u
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Arithmetic and Geometric Sequences An ARITHMETIC SEQUENCE is one in which the difference between successive terms is constant, such as 2, 4, 6, 8,… Here the common difference is 2 A GEOMETRIC SEQUENCE is one in which the ratio of successive terms is a constant such as ½, 1, 2, 4, 8.. Here the common ratio is 2
Image of page 6
Geometric series What is the sum of the first n terms in a geometric sequence? Assume the common ratio r 1 S n = a + ar + ar 2 + ar 3 + … + ar n- 1 = a (1 – r n )/(1 – r ) Proof (Workbook, Chapter 3, Sect. 2.5) S n = a + ar + ar 2 + ar 3 + … + ar n- 1 rS n = ar + ar 2 + ar 3 + … + ar n- 1 + ar n S n rS n = a – ar n which then gives the formula
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Application: finance You have £500 in the bank and the interest rate is 6% (i.e. 0.06) per year. Interest is paid at the end of the year. At the end of year 1 you receive 0.06 x 500 = £30 The interest is added to the £500, so you have £530 At the end of year 2 you receive 0.06 x £530 = £31.80, so your total sum is £561.80 Generally: if you invest P and interest is paid annually at interest rate i then after one year you have P (1 + i ) After two years you have P (1 + i )(1 + i ) = P (1 + i ) 2 After t years you have P (1 + i ) t The amount of money in your bank at the end of each year is a geometric sequence with common ratio (1 + i )
Image of page 8
How much will pension contributions be worth?
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern