MATH 1005 Study Guide - Final Guide: Fourier Series, Periodic Function, Absolute Convergence
4 Jul 2017
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Document Summary
Given a function f (x), the taylor series of f (x) centred at a is where. Xn=0 cn(x a)n cn = f (n)(a) n! (here f (n)(a) is the nth derivative of f evaluated at a). The taylor series for f (x) may or may not converge to f (x) (we didn"t see enough in class to decide when f (x) = t (x)). The radius of convergence of t (x) is given by n!1 cn cn+1 R = lim if the limit exists (if the limit doesn"t exist, the ratio test can be used to nd r). Nd the interval of convergence, these endpoints must be tested, and either included or excluded from the interval. Some important taylor and maclaurin series that we saw include: ex = sin(x) = cos(x) = 3! (radius of convergence is r = ) ( 1)nx2n+1 (2n + 1)! 7! (radius of convergence is r = ) ( 1)nx2n (2n)!
