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(9 points) Give examples of the following or explain why they cannot not exist. a. A power series which converges for no values of x. b. A non-increasing sequence diverging to 00. c. A divergent series for which the Root Test is inconclusive. (35 points) Determine whether the following sequences converge, and if so, whether that convergence is absolute or conditional. Evaluate the series if you can. You may use each test for convergence] divergence exactly once. For your own purposes, you may use whatever you please, but to receive full credit, you must use different tests for different series. 1 3- 2:10:11 \"244' b 2:312 16 x (0.6)\". C- 2:10:05 d 2;;2 1101111)? e- 2110:21121-1. f. macaw-\"2. g. 2:;0214x(12)n. (7 points) Verify 0.9 = 2:0 ('35) (14 points) Let f(x) = 92" + cos x. a. Compute T400 for f(x) centered at c = 1r. b. Give the Taylor series for f(x) centered at c = 7t and prove that its radius of convergence is 00. (10 points) Use Taylor's Theorem to prove that the Alternating Harmonic Series converges to In 2. (10 points) We encountered useful inequalities in this unit. a. Use the Triangle Inequality to prove that |2$f=o xnl s Eigolxnl. This is not trivial. b. Verify that if 235:0 x3, and 2310 y converge, then 2,10%)!" converges and that convergence is absolute. (15 points) Recall the Hermite polynomials. We verified that Hn(t) satisfies 3:" tx' = nx. Use power series to compute H, (t) and H30). Hint: an = 1 and an\" = 0 for Hn(t). Extra Credit: We define an infinite product as the limit of partial products: H320 xn = 1 lim N. x = limx x--->