Lecture 24 (Power Series and Interval of Convergence)

Lecture 24 (Power Series and Interval of Convergence) -...

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Unformatted text preview: Monday, March 5 Lecture 24 : Power Series and their interval of convergence. (Refers to Section 8.6 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Define a power series centered at a , determine both the radius of convergence and the interval of convergence of a power series, express a power series which is a relative of a geometric series as a function. 24.1 Definition A power series centered at a is a series of the form with a variable x . We make a few remarks about this concept: - This series differs from the series we previously studied by the fact that it contains a variable x . - This means that at certain values of x the series may converge to a particular number. The number it converges to depends on the value we give to x . At still other values of x the power series may diverge and so is not equal to any number. This gives the impression that a power series is a function . We refer to set of numbers on which series converges as its domain . - We will be interested in finding an elementary function f ( x ) which represents the series. 24.1.1 Observations Students are already familiar with a power series, even though we didn't refer to it as such. We know that the series converges to 1/(1 r ) for 1 < r < 1, including the number r = 0, while it diverges elsewhere. - We might prefer to say that the power series, j = 0 to x j , centered at 0, converges to the function 1/(1 x ) on the interval ( 1, 1) while it diverges elsewhere. We might even want to write 24.2 Examples Here are a few examples of power series centered at : Our study of power series will focus on two questions: 1. For what values of x does a power series converge? 2. To what function does it converge to? 24.3 Theorem A power series centered at a , 1. converges at least at x = a . It may converge for all x , in which case it converges absolutely for all real numbers. If it does not converge for all real numbers then must converge on some interval ( a R , a + R ), ( a R , a + R ] , [ a R , a + R ) or [ a R , a + R ], where R is some positive number, ( i.e., on | x a | < R ), while it diverges on | x a | > R ....
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Lecture 24 (Power Series and Interval of Convergence) -...

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