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Analysis I Oleg Zaboronski Assignment 8 Due Monday 26 November 15:00 (return to your supervisor’s pigeon hole) Problem 1. Use the Comparison Test to determine whether each of the following series converges or diverges. ∑∞ √ 1 (a) n=1 3 2 n +1 (b) ∑∞ (c) ∑∞ n=1 n=1 ( 1 √ 3 7 n +1 √ n+1− √ n) Problem 2. Determine whether each of the following series converges or diverges. Make your reasoning clear. ∑∞ 1 (a) n=1 n(n+1)(n+2) (b) ∑∞ 5n +4n n=1 7n −2n Problem 3. Let an = n2 2n . Prove that if n ≥ 3, then 8 an+1 ≤ . an 9 By using this inequality for n = 3, 4, 5, ..., prove that ( )n 8 an+3 ≤ a3 . 9 Using the Comparison Test and ∑∞results concerning the convergence of the Geometric Series, show ∑ that n=1 an+3 is convergent. Now use the Shift Rule to show that ∞ n=1 an is convergent. Problem 4. Write down an example of a convergent series and a divergent series both of which satisfy the condition an+1 an → 1. Problem 5. Use the Ratio Test to determine whether each of the following series converges or diverges. your reasoning ∑ ∑∞Make ∑∞ clear. 2n 3n n! (a) ∞ (b) (c) n n=1 n! n=1 n n=1 nn/2 ∑ 1 1 1 1 Problem 6. Show that 200 n=101 k = 101 + 102 + ... + 200 ∈ [0.688, 0.694]. ∑ 1 Problem 7. Show that (a) ∞ n=1 (n+1) log(n+1) is divergent; ∑∞ 1 (b) n=1 (n+1)(log(n+1)) 2 is convergent.