Question

Question 1 3+cos(n) 2n X Which of the following properties hold for the sequence an for n 2 1? l. Bounded Il. Monotonic IIl. Convergent Selected Answer a. I only a. I only b. Il only c. I and Il only d. I and Ill only e. I, II, and III Remember what these conditions mean: Bounded means all terms of the sequence have to lie within a specific range of values. Monotonic means the sequence is ALWAYS increasing or ALWAYS decreasing. It can never increase from one term to the next, then decrease on the next term. Convergent means that the limit of the sequence exists. Answers: Response Feedback: Question 2 What can we determine by applying the Test for Divergence (and no other tests) to the series Selected Answer d. The series diverges. Answers: a. The series converges, but we don't know its sum. b. The series converges to 0. C. The series converges to d. The series diverges. e. The test is inconclusive (i.e. we can't determine anything).

Answer 1

Let a_n = 3 + cos(n)/2n now -1 lessthanorequalto cos(n) lessthanorequalto 1 for all n -1 + 3 lessthanorequalto 3 + cos(n) lessthanorequalto 3 + 1 2 lessthanorequalto 3 + cos(n) lessthanorequalto 4 2/2n lessthanorequalto 3 + cos(n)/2n lessthanorequalto 4/2n 1/n lessthanorequalto a_n lessthanorequalto 2/n for all n a_n is bounded Also a_1 = 3 + cos 1/2 = 3 270 (approx) a_2 = 3 + cos 2/4 = 2 896 a_3 = 3 + cos 3/6 = 2 835 a_4 = 3 + cos 4/8 = 2 918 a_3 = a_2 < a_1 and a_3 > a_4 a_n is not monotonic lim_x rightarrow infinity a_n = lim_x rightarrow infinity 3 + cos(n)/2n = 0 a_n is convergent hence I, III are correct and II is incorrect option (D) I and III only