fall17mth143.practice2.4-ConvergenceTestsII.pdf

Absolutely convergent conditionally convergent

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? absolutely convergent conditionally convergent divergent (iv) a < 0 < b ≤ - a ? ? absolutely convergent conditionally convergent divergent Answer(s) submitted: (incorrect) Correct Answers: divergent absolutely convergent absolutely convergent divergent 16. (1 point) Consider the series n = 1 c n , where c n = n p a n b n , with b 6 = 0. Complete the following: (i) Compute lim n | c n + 1 c n | in terms of p , a , and b . lim n | c n + 1 c n | = (ii) Based on your answer to (i), the series converges abso- lutely by the ratio test if a belongs to what range of numbers (your answer should be in terms of p and b )? The series converges with a on help (inter- vals) (iii) Based on your answer to (i), the series diverges by the ratio test if a belongs to what range of numbers (your answer should be in terms of p and b )? The series diverges with a on help (inter- vals) (iv) Based on your answer to (i), the ratio test does not tell if this series converges if a belongs to what range of numbers (your answer should be in terms of p and b )? The ratio test does not give us convergence information with a on help (intervals) Answer(s) submitted: (incorrect) Correct Answers: |a|/|b| (-|b|,|b|) (-infinity,-|b|) U (|b|,infinity) {-|b|} U {|b|} 17. (1 point) Select the FIRST correct reason why the given series converges. A. Convergent geometric series B. Convergent p series C. Comparison (or Limit Comparison) with a geometric or p series D. Alternating Series Test E. None of the above 1. n = 1 ( n + 1 )( 99 ) n 10 2 n 2. n = 1 7 ( 4 ) n 7 2 n 3. n = 1 ( - 1 ) n 7 n + 7 4. n = 1 sin 2 ( 3 n ) n 2 5. n = 1 cos ( n π ) ln ( 2 n ) 6. n = 1 ( - 1 ) n ln ( e n ) n 4 cos ( n π ) Answer(s) submitted: 6
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(incorrect) Correct Answers: C A D C D B 18. (1 point) (a) Check all of the following that are true for the series n = 1 6 n 2 n - 1 A. This series converges B. This series diverges C. The integral test can be used to determine conver- gence of this series. D. The comparison test can be used to determine con- vergence of this series. E. The limit comparison test can be used to determine convergence of this series. F. The ratio test can be used to determine convergence of this series. G. The alternating series test can be used to determine convergence of this series. (b) Check all of the following that are true for the series n = 1 ln ( 2 n )+ 6 n n 2 A. This series converges B. This series diverges C. The integral test can be used to determine conver- gence of this series. D. The comparison test can be used to determine con- vergence of this series. E. The limit comparison test can be used to determine convergence of this series. F. The ratio test can be used to determine convergence of this series. G. The alternating series test can be used to determine convergence of this series. Solution: SOLUTION (a) For this series, the terms in the series involve exponen- tials, so that the ratio test is a good choice to test convergence. If we’re careful we can find a comparison series that will work, and the limit comparison test will definitly work. The require- ments for the integral test are satisfied, but we can’t integrate the function f ( x ) = 6 x 2 x - 1 , so it’s not good to assess the convergence of the series. The alternating series test isn’t applicable, because
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