lect118_26_w13 - Wednesday March 13 Lecture 26 Expressing...

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Wednesday, March 13 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. 26.1Questions In previous examples we saw that the geometric series Σj=0 to x jconverges 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 ofx. The theorem below shows that derivatives and the integrals of functions which are relatives of 1/(1 x) are functions

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