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Lecture 27 (Taylor and Maclaurin Series Generated by a Function)

# Lecture 27 (Taylor and Maclaurin Series Generated by a Function)

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Monday, March 12 Lecture 27 : Taylor series and Maclaurin series generated by a function f ( x). (Refers to Section 8.8 your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Define a Taylor series centered at a which is generated by a function f ( x ), define a Maclaurin series generated by a function f ( x ), find the Taylor series centered at a which is generated by a function f ( x ) and determine its radius of convergence. 27.1 Definition Let f ( x ) be a function which is differentiable an infinite number of times on an open interval centered at a . The series is called a Taylor series centered at a generated by the function f . We say that the function f ( x ) generates the series since its coefficients are obtained from a mechanism which involves the function f ( x ). If a = 0 then f ( x ) generates the Taylor series centered at 0. A Taylor series centered at 0 has the special name Maclaurin series . Note that f (0) ( x ) is defined as being f ( x ). 27.1.1 Example Show that Maclaurin series generated by the function f ( x ) = e x is and determine its radius of convergence. We first obtain f ( n ) (0) for all n : ( e x ) ( n ) | x = 0 = e x | x = 0 = 1 for all n . Thus f ( n ) (0) / n ! = 1/ n !.

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Hence the Maclaurin series which is generated by f ( x ) = e x is We now find it radius of convergence: Thus the series converges for all values of x . 27. 2
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Lecture 27 (Taylor and Maclaurin Series Generated by a Function)

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