# lect118_27_w13 - Friday March 15 Lecture 27 Taylor series...

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Friday, March 15 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 awhich is generated by a functionf(x), define a Maclaurin series generated by a functionf(x), find the Taylor series centered at awhich 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 ata. The series is called a Taylor series centered at a generated by the function f . We say that the function f (x) generatesthe 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) =exis and determine its radius of convergence.