1. Given the sequence definitions, answer the following: a) an - 3...

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1. Given the sequence definitions, answer the following: a) an - 3n+2 , write out the first three terms. 4n-1 a1 = a2 - a3 = Does the sequence converges or diverges? If it converges, tell what value it converges to b) b1 = 2,b2 - 1.6n - bn-1 + write out the first three terms. b1 = b2 = b3 = Does the sequence converges or diverges? If it converges, tell what value it converges to. 2. Does the improper integral 500 - 2 da converge? Show your work 3. Find the sum of the following series if it is converges, otherwise state that it diverges. a) 8 3n+(-1)n n=1 4n2 b) \ 80 4 n=2 n2 - 1 4. State whether (You do not have to prove) the following statement are TRUE(T) or FALSE (F). If it is false give an counterexample. a) For the series [n-1 an. if limn too an = 0. then the series converge. b) Alternating series always converges. c) If a series converges then the series formed by taking the absolute values of the terms is guaranteed to converge. d) For the integral test to work, the corresponding function should be increasing eventually. e) Any series that can be tested for convergence using the Direct Comparison Test, can always be tested using the Limit Comparison Test. 5. Use the Direct Comparison Test (DCT) or the Limit Comparison Test (LCT) as appropriate to test the convergence of the following series: a) 8 In n n n 80 n² +sin n b) n=. 1 3n7 +5n-1 6. Use the Alternating Series Test (AST) or any other test to check the convergence of the following S a) [n-3 (-1)" n 7n2- 5 21n² +n b) [no,1 (-1)n-1 n n2+4 7. Use the Integral Test to check whether the following series converge. Show you work. 00 Ln-1 4 Extra Credit Problem (8 points): Only answer any two parts. Test whether the following series are convergent. Mention the test you use and clearly state your conclusion. 12 80 a) n=1 (1 - 3 k b) 00 k=1 2020 (-)) n=5 8 3 cos(3n)+2 2n 00 4rn-5n+3 tan n=] 12n°+7n2-2020

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