Mathematics in economics Lecture 9 Mgr. Jiří Mazurek, Ph.D. Mathematics in Economics/PMAT Lecture 9 Geometric series and its application • • 1. Geometric series belong among the most simple, yet the most used series in practice. Definition: Simply put, elements of a geometric series is generated by multiplying each element by a constant q called the quotient. Lecture 9 Geometric series – numerical examples Examples of geometric series: In this series a1 = 2 and q = 2/3. Another example: Here, a1 = 1 and q = 2. • • 1. Lecture 9 Geometric series and its sum • • 1. The sum of first n numbers of a geometric series is given as: The sum of a convergent infinite geometric series is given as: Note: The latter formula is obtained as a limit to infinity of the former formula. Lecture 9 Geometric series and its convergence • • 1. Geometric series is convergent if and only if: Only in this case it is guaranteed elements of a series are approaching 0 as n approaches infinity. Also, the formula for the sum can be used only for convergent series! Lecture 9 Geometric series – Problem 1 • • 1. Decide convergence and find the sum of the series: We see that a1 = 1 and q = 1/3. The series is convergent. Its sum: Lecture 9 Geometric series – Problem 2 • • 1. Decide convergence and find the sum of the series: We see that a1 = 2/5 and also q = 2/5. The series is convergent. Its sum: Lecture 9 Geometric series – Problem 3 • • 1. Decide convergence and find the sum of the series: We see that a1 = 4/3 and also q = 4/3. The series is divergent! Its sum is infinity. We cannot use the formula for its sum! It is a common mistake in tests! Lecture 9 Geometric series – Problem 4 • • 1. Decide convergence and find the sum of the series: We see that a1 = 1 (!) and q = 3/4. The series is convergent. Its sum: Lecture 9 Geometric series and its application in economics • • 1. A total revenue of a book or a movie: when a new book/movie is released, the highest revenue comes from the first week, and then the revenue decreases usually by 30-40%. This situation can be modelled by an infinte geometric series, which gives a very good approximation of real revenues. Lecture 9 Geometric series and its application in economics Example: Find the total revenue of a movie which earns 70 mil dollar during the first week, while earnings decrease by 30 % each week. Solution: We have a geometric series Clearly, a1 = 70 and q = 0.7. • • 1. Lecture 9 Geometric series and its application in economics Example: Find the total revenue of a book which earns 0,5 mil dollar during the first week, while earnings decrease by 20 % each week. Solution: We have a geometric series Clearly, a1 = 0.5 and q = 0.8. • • 1. Lecture 9 Geometric series and its application • • 1. Other applications of geometric series include present values of annuities, bonds, etc. Generalization of number series is function series, which will be dealt in the second part of this lecture and Lecture 10. Lecture 9 Geometric series – Problems to solve 1 • • 1. Is the series a geometric series? If yes, find its sum: Lecture 9 Geometric series – Problems to solve 2 • • 1. Is the series a geometric series? If yes, find its sum: Lecture 9 Geometric series – Problems to solve 3 • • 1. A movie earns 45 million dollars on its opening week. Find the total revenue of the movie for a decrease of: a)20% b)30% c)40% d)50% weekly. e) Lecture 9 Function series – an introduction • • 1. In the second part of this lecture we will introduce function series, which are a generalization of number series. By a function series we mean the following series: Lecture 9 Function series – an introduction • • 1. The main goal is to find x for which a function series converges. This set is called range of convergence. Example: Let be a given series. Now, let x = ½. Then, we obtain: This series is convergent. However, when x = 2, we get: which is divergent. Lecture 9 Function series Power series • • 1. The simplest function series are power series: Power series converge on an interval , where a is called a centre of a series and is a radius of a convergence. Lecture 9 Function series Power series – cont. • • 1. The radius of convergence is given by the following formulas: A convergence of a series in points a + ρ and a – ρ must be resolved separately. Lecture 9 Function series Power series – Problem 1 • • 1. Find the interval of convergence of the series . Solution: We see that a = 2 and . Hence: It can be shown the series is divergent at -1 and +1. Lecture 9 Function series Power series – Problem 2 • • 1. Find the interval of convergence of the series . Solution: We see that a = -5 and . Therefore, the series is convergent only at its center (-5). Lecture 9 Function series Power series – Problem 3 • • 1. Find the interval of convergence of the series . Solution: We see that a = -1 and . Hence: Lecture 9 Function series Power series – Problem to solve • • 1. Find the interval of convergence of the series: Lecture 9 • • 1. Thank you for your attention