REVIEW3 - momin (rrm497) Review 3 cheng (58520) 1 This...

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Unformatted text preview: momin (rrm497) Review 3 cheng (58520) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine whether the sequence { a n } con- verges or diverges when a n = ( 1) n 1 n n 2 + 5 , and if it converges, find the limit. 1. converges with limit = 1 5 2. converges with limit = 0 correct 3. converges with limit = 1 5 4. converges with limit = 5 5. sequence diverges 6. converges with limit = 5 Explanation: After division, a n = ( 1) n 1 n n 2 + 5 = ( 1) n 1 n + 5 n . Consequently, | a n | = 1 n + 5 n 1 n . But 1 /n 0 as n , so by the Squeeze theorem, lim n | a n | = 0 . But | a n | a n | a n | , so by the Squeeze theorem again the given sequence { a n } converges and has limit = 0 . keywords: 002 10.0 points Determine if the sequence { a n } converges when a n = 7 n 2 (2 n 1)! (2 n + 1)! , and if it converges, find the limit. 1. converges with limit = 7 2 2. converges with limit = 2 7 3. converges with limit = 4 7 4. does not converge 5. converges with limit = 7 4 correct Explanation: By definition, m ! is the product m ! = 1 2 3 . . . m of the first m positive integers. When m = 2 n 1, therefore, (2 n 1)! = 1 2 3 . . . (2 n 1) , while (2 n + 1)! = 1 2 3 . . . (2 n 1)2 n (2 n + 1) . when m = 2 n + 1. But then, 7 n 2 (2 n 1)! (2 n + 1)! = 7 n 2 2 n (2 n + 1) 7 4 as n . Consequently, the given sequence converges with limit = 7 4 . momin (rrm497) Review 3 cheng (58520) 2 003 10.0 points Determine whether the infinite series summationdisplay n =1 ( n + 1) 2 n ( n + 2) converges or diverges, and if converges, find its sum. 1. converges with sum = 1 4 2. converges with sum = 1 3. converges with sum = 1 2 4. diverges correct 5. converges with sum = 1 8 Explanation: By the Divergence Test, an infinite series n a n diverges when lim n a n negationslash = 0 . Now, for the given series, a n = ( n + 1) 2 n ( n + 2) = n 2 + 2 n + 1 n 2 + 2 n . But then, lim n a n = 1 negationslash = 0 . Consequently, the Divergence Test says that the given series diverges . keywords: infinite series, Divergence Test, ra- tional function 004 10.0 points Determine whether the series summationdisplay n = 3 ( 1) n +2 5 n ln n converges conditionally, converges absolutely, or diverges. 1. series converges absolutely 2. series converges conditionally correct 3. series diverges Explanation: The given series can be rewritten as summationdisplay n =3 ( 1) n +2 5 n ln n = 5 summationdisplay n =3 ( 1) n f ( n ) , where f ( x ) = 1 x ln x = ( x ln x ) 1 ....
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REVIEW3 - momin (rrm497) Review 3 cheng (58520) 1 This...

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