MTH 1322 – Calculus II Homework 16 – Power Series 1. Find the radius and interval of convergence of the power series (b) ∞∑ n=1 nxn (e) ∞∑ n=1 (x+ 1)n n 3n 2. Expand f(x) as a power series and find...

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MTH 1322 – Calculus II Homework 16 – Power Series 1. Find the radius and interval of convergence of the power series (b) ∞∑ n=1 nxn (e) ∞∑ n=1 (x+ 1)n n 3n 2. Expand f(x) as a power series and find the interval on which the expansion is valid. (b) f(x) = 1 1 + 2x 3. Find the Taylor series of f(x) centered at x = a. (c) f(x) = sin x, a = π 2 6. Find the Maclaurin series of f(x) and state the largest open interval on which the series expansion is valid. You may start with a Maclaurin series of a function you memorized. (b) f(x) = cos(2x) (d) f(x) = x2e−x (f) f(x) = tan−1 x Hint : d dx tan−1 x = 1 1 + x2 7. If f(x) = ∞∑ n=0 (−1)n x 2n (2n+ 1)! for all x, find f (6)(0). 8. Approximate the value of the definite integral correct to within 10−3. (d) ∫ 1 0 e−x 2/2 dx