MATH 1272 Lecture Notes - Lecture 1: Taylor Series, If And Only If
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Assume f(x) has a power series about a with radius of convergence r. so. Definition- the taylor series of f centered at a. If a=0, we call it a maclaurin series. We have only shown that if f can be represented by a power series, then f(x)= it"s taylor series when |x-a| < r. There are cases where we can construct a taylor series, but it is not equal to our function. Construct a taylor series for f(x)=|x| centered at a=1. So |x) its taylor series when x<0. Make another table to find what power -1 is raised to. Sin(2x) has a maclaurin series (centered at 0) But |x| x for x > 0. Recall the partial sums of a power series. The nth taylor polynomial tn(x) of f at a is the nth partial sum of the series. Note that the nth taylor polynomial has degree n. Suppose f is a function with taylor series.
