Assignment+9.pdf - Mathematics UN 1102 Section 1 Assignment...

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Unformatted text preview: Mathematics UN 1102, Section 1: Assignment 8 Fall 2018 Due before class on Wednesday 11.28.2018 Before doing this problem set review Sections 11.4  11.7 from the book. Problem I. For each of the following series determine if they are absolutely convergent, conditionally convergent or divergent. Write down one sentence explaining how you concluded which case holds. Here are some examples of acceptable answers: P∞ P∞ 1 1 is absolutely convergent by the Comparison test with • 2 n=1 n +3n+5 n=1 n2 (−1)n n=1 ln(n+1) P∞ is conditionally convergent by the Alternating series test and not absoP 1 lutely convergent by comparison with ∞ n=1 n P P∞ 1 • Since limn→∞ sin(1/n) = 1, and ∞ n=1 n = ∞ we conclude that n=1 sin(1/n) is diver1/n gent by the Limit Comparison Theorem. P 2n • The series ∞ n=1 n! is convergent by the Ratio test with L = 0. h in P 2n2 +5 − • The series ∞ is absolutely convergent by the Root test with L = 2/3. n=1 3n2 +4n+5 • • The series P∞ • The series P∞ 1 n=2 n ln(n) n=1 is divergent by the Integral test with function (−1)n n n+2 1 . x ln(x) is divergent by the Divergence test. Essentially, this is a true false type of question, with a bit of justication. The following are all exercises from Section 11.7 in the book. P P∞ n−1 P∞ P∞ P∞ en n2 −1 n n2 −1 n n2 −1 1. (a) ∞ n=1 n3 +1 (c) n=1 (−1) n3 +1 (d) n=1 (−1) n2 +1 (e) n=1 n2 n=1 n3 +1 (b) 2. (a) P∞ (b) P∞ 3. (a) P∞ n2 e−n (b) P∞ 4. (a) P∞ 5. (a) P∞ 6. (a) P∞ tan(1/n) (b) 7. (a) P∞ (b) n2n n=1 (1+n)3n n=1 3 2k−1 3k+1 k=1 kk n=1 (−1) n=1 e1/n n=1 n2 n=1 (b) n ln(n) √ n √1 n=1 n ln(n) 1 n3 1 3n √ P∞ n4 +1 n=1 n3 +n (b) P∞  (c) 1/3 −1 k√ k=1 k( k+1) P∞ n=1 P∞ n−1 n4 n=1 (−1) 4n P∞ (c) (c) (d) (d) √1 k=1 k k2 +1 P∞ 1·3·5····(2n−1) n=1 2·5·8····(3n−1) (c) P∞ n sin(1/n) (c) (−1)n n=1 en +e−n P∞ + (c) n=1 (−1) P∞ n! n=1 en2 √ n n n=1 (−1) n+5 P∞ 1 n P∞ n π 2n n=1 (−1) (2n)! P∞ (d) n=1 3n n2 n! P∞ n=2 P∞ P∞ (e) (d) n=1 P∞ 1 k=1 2+sin(k) n2 +1 5n 5n n=1 3n +4n sin(2n) n=1 1+2n (−1)n−1 √ n−1 cos(1/n2 ) (d) (d) P∞ (e) (e) P∞ k ln(k) k=1 (k+1)3 (n!)n n=1 n4n P∞ 8. (a) P∞ 9. (a) P∞ n=1 n2 n n+1 (b) P∞ 1 n=1 n+n cos2 (n) (c) √ P∞ √ n n n ( 2 − 1) (b) n=1 n=1 ( 2 − 1) 2 P∞ 1 n=1 ln nln n (d) P∞ 1 n=1 n1+1/n ...
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