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MATH 1320 : Spring 2014 Lab 4 Lab Instructor : Kurt VanNess Name: Score: Write all your solutions on a separate sheet of paper. 1. Suppose you know that a sequence {an } is an increasing sequence and all its terms lie between the numbers 2 and 4. Explain why the sequence must have a limit? What can you say about the value of the limit? 2. Find the limit of the sequence ( 3. Suppose that ∞ X √ q 2, √ 2 2, r q ) √ 2 2 2, . . . . an (an 6= 0) is known to be a convergent series. Prove that n=1 ∞ X 1 is a divergent a n=1 n series. P P P 4. If an and bn are both divergent, is (an + bn ) necessarily divergent? P∞ P∞ 5. If the nth partial sum of a series n=1 an is sn = n−1 n=1 an . n+1 , then find an and 6. Compute the sum of the series: ∞ X 2 . [Hint: Telescoping - Use partial fraction decomposition] n(n + 2) n=1 7. Determine whether the following series converge or not by using either the Integral Test or the Comparison Test: ∞ ∞ X X 1 n √ , . 3 5+n+1 n(ln n) n n=2 n=1 8. (Alternating Series) (a) Show that the series ∞ X (−1)n converges. n2n n=1 k X (−1)n . Then S(k) is the partial sum of the series. Compute k so n2n n=1 that S(k) is within 0.01 of the sum of the series. (b) Let S(k) be defined by: Page 1 of 1