a. c. e. b. d. a. c. e. b. d. Solution use the same approach Find the Maclaurin

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use the same approach Find the Maclaurin series for f(x)=(1-x)^-2 using the definition of a Macluarin series,[ Assume that f has a power series expansion. Do not show that Rn(x) ->0.] Also find the associated radius of convergence Solution: Gee, that sounds more like a command than a question! The Maclaurin series is f(x) = f(0) + [f ' (0) x / 1!] + [f ' '(0) x^2 / 2! ] + ... + [ f^[n](0) x^n / n! ] + ... So find the first few derivatives f ' (x) = -2(1 - x)^-3 * -1 = 2(1-x)^-3 f ' ' (x) = -6(1-x)^-4 * -1 = 6(1-x)^-4 f ' ' ' (x) = 24(1-x)^-5 etc. When you plug x=0 into all of these, you get the coefficient times 1. So in general, the nth derivative is (n+1)! when you plug in x=0. So the series is: f(x) = f(0) + [f ' (0) x / 1!] + [f ' '(0) x^2 / 2! ] + ... + [ f^[n](0) x^n / n! ] + ... f(x) = 1 + [2! x / 1!] + [3! x^2 / 2! ] + ... + [ (n+1)! x^n / n! ] + ... f(x) = 1 + 2x + 3x^2 + ... + (n+1)x^n + ...

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