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### Lecture 26 (Expressing Functions as a Power Series)

Course: MATH 118, Winter 2012
School: Waterloo
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Word Count: 485

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March Wednesday, 7 Lecture 26 : Expressing functions as a power series: Derivatives and Integrals of power series. (Refers to Sections 8.6 and 8.7 ) After having practiced the problems associated to the concepts of this lecture the student should be able to: Express a function which is a relative of the geometric function 1/(1 x) as a power series. 26.1 Questions In previous examples we saw that the geometric...

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March Wednesday, 7 Lecture 26 : Expressing functions as a power series: Derivatives and Integrals of power series. (Refers to Sections 8.6 and 8.7 ) After having practiced the problems associated to the concepts of this lecture the student should be able to: Express a function which is a relative of the geometric function 1/(1 x) as a power series. 26.1 Questions In previous examples we saw that the geometric series j=0 to x j converges to the function 1/(1 x) on the interval (1, 1). We can then also say that the function 1/(1 x) has a power series representation. We wonder if there are other functions which have a power series representation. In this lecture we will see that there are many. In particular those who, in some way, are related to the function 1/(1 x). Look carefully at the following examples to see this: Observe that the above equalities only hold for certain values of x. The theorem below shows that derivatives and the integrals of functions which are relatives of 1/(1 x) are functions for which we can also obtain power series representations. Below we will work with functions which are relatives of the geometric function 1/(1 x) by differentiation and integration. 26.1.1 Example Find a power series representation of the function 1/(1 x)2. Solution: We will make use of the form () = (), for any Note that in the chain of equalities below we must restrict x to x < 1. 26.1.2 Example Find a power series for the function f(x) = ln(1 + x) and give its interval convergence. Solution: Recall of that () () = () To find the interval of convergence we must test convergence at x = 1 and x = 1. - - The function ln(1 + x) is not defined at x = 1 so we need only test for convergence at x = 1. When x = 1, this series is the alternating harmonic series which is known to converge. So the series converges to ln(1 + x) on the interval (1, 1] 26.2 Example Express the function ln x as a power series. (Note that (d/dx) ln x = 1 /x.) 26.3 Example Show that f(x) = arctan x can be expressed as the power series Verify that its radius of convergence is R = 1. We only outline the solution by setting up the integral: 26.4 Example Express the function f(x) = 1/(x 2) as a power series centered at 3 and state the radius of convergence as well as the interval of convergence. Radius of convergence R = 1, and interval of convergence is (2, 4). 25.5 Example Express the function as a power series centered at 4 and state the radius of convergence as well as the interval of convergence. Note: The function 2x2 5x + 3 factors to (x 1)(2x 3) and see that f (x) the expression can be broken up in partial fractions as follows: Solution: 26.6 Example Express the function as a power series centered at 0 and state the radius of convergence as well as the interval of convergence. 26.7 Example Find a power series representation for the function centered at 0 and state the radius of convergence as well as the interval of convergence. Solution:
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