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[Must prove results from scratch] 1) Suppose Pāˆž n=0 anz n is a power series with radius of convergence R > 0. Show that the function is f(z) = Pāˆž n=0 anz n is analytic in its disk of convergence. 2) Suppose f is analytic in an open disk D = {z āˆˆ C : |z| < R}. State and prove the Taylor theorem for this f. (namely in the case z0 = 0). 3) Give a counter-example in R that a function f : R ā†’ R has derivatives of all orders, but its Taylor series does not equal to f.

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