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Problem 1

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Problem 1. Determine whether the series below converges or diverges. Please state which test you are using and explain your reasoning as usual. 
100"1-1 E 3 27n n=3. 
c's (-1)k Problem 2. Determine whether the series E  kw/4 absolutely converges, conditionally k=0 
converges or diverges. Justify your work. Problem 3. Consider the series 1 + Ex s Lr2 x2 + Lr3 033 s 2 3 4 
(a) Write the series in summation notation. 
(b) Determine the interval of convergence and the radius of convergence for the series. 
2 cos(2x) — 2 4x2 Problem 4. Compute lira  z-40 2x4 Problem 5. Let f be a function that has derivatives of all orders at x = 2. Let Pn(x) denote the nth degree Taylor polynomial for f centered about x = 2. You are given 1(2) = 2, f" (2) = —1, fill (2) = 0 and f(4)(2) = 3. Suppose P4 (4) = —4. Find P4 (X)• 
00 10n + 1  Problem 6. Suppose a series E a„ has partial sums sn defined by sn = n 
n=1 
(a) Compute E an. Show all work. n=1 
(b) What is an? Explain your reasoning.