3. [8 points] Let a1, a2, a3, ... be a sequence of positive numbers such that the power series Can (2 + 1 ) " 7 = 0 has a radius of convergence R = NICT (a) On what (open) interval does the series > an(x + 1)" converge? n= 0 (b) Does the series an converge or diverge? Justify your answer. n=0 (c) Determine lim an (you may reference your answer to part (b)). Briefly justify your answer.(d) Does the sequence In(a1), In(a2), In(a3), ..., In(an), ... converge or diverge? Justify your answer. (e) Let f(x) = Can(x + 1)". Find f"(x) and state where it converges. (f) Does the series > n(n - 1)an . 2"-2 converge or diverge? Justify your answer.2. [5 points] m (a) Argue that the integral I 005$\" 1 t3/2 answer (indicating any test or theorem that you are using as well as checking any required conditions). dt converges Be sure to use correct notation and to fully justify your m En (b) Since /1. lofsgmdt converges, you may also conclude that /1. {3:3}? convergence test for type I integrals). Use this fact and integration by parts to argue that the integral dt converges [by the absolute converges