Consider the function f(x) = (8)3. We want to find the Taylor series of f(x) at x=7 and at x = -5. (a) The nth derivative of f(x) is f(n) (x) = (b) Atx=7, we get f(n) (7) = The Taylor series at x=7 is +00 Τ(x) = Σ n=0 To find the radius of convergence, we use the ratio test. an+1 L = lim 8+←# an and so its radius of convergence is R = Enter INF if the radius is infinite. (c) Atx=-5, we get f(n) (-5)= The Taylor series at x=-5 is Τ(α) = Σ n=0 To find the radius of convergence, we use the ratio test. an+1 L = lim 8+← an and so its radius of convergence is R = Enter INF if the radius is infinite. (x-7)" |x-7 (x+5)" |x+51

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