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Free Respоnse Wоrk аll fоur questions below on your own pаper, showing аll steps. Label each question and part clearly. Show all work, including all key steps. Answers without supporting work receive no credit. Your work must be legible enough for me to read and follow your reasoning. Use arrows or other notation if needed to make the order of your work clear. FR1. (20 points) For each series below, name the convergence test you will use, show your work, and conclude whether the series diverges, converges absolutely, or converges conditionally. (a) (5 points) (displaystylesum_{n=1}^{infty}frac{3^n}{n!}) (b) (7 points) (displaystylesum_{n=1}^{infty}frac{n+1}{n^3+2}) (c) (8 points) (displaystylesum_{n=1}^{infty}(-1)^n,frac{n}{n^2+1}) FR2. (8 points) (a) (5 points) Find the third degree Taylor polynomial for (f(x)=sin x) centered at (a=dfrac{pi}{6}). (b) (3 points) Explain why you cannot use the Alternating Series Estimation Theorem to find the maximum possible error of using your answer from part (a) to approximate (sin 1). FR3. (12 points) (a) (5 points) Find the power series representation for (f(x)=dfrac{1}{6-x}) centered at (a=0). (b) (5 points) Use your answer from part (a) to find the power series representation for (g(x)=dfrac{1}{(6-x)^2}) centered at (a=0). (c) (2 points) Use your answer from part (b) to find the power series representation for (h(x)=dfrac{x^2}{(6-x)^2}) centered at (a=0). FR4. (10 points) Consider the curve given by the parametric equations (x(t)=t^2-4t, qquad y(t)=t^3-3t.) (a) (5 points) Find all values of (t) at which the curve has a vertical tangent line, and give an equation for each such line. (b) (5 points) Find all values of (t) at which the curve has a horizontal tangent line, and give an equation for each such line.
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