cal14 - choi(kc24547 – HW14 – BERG –(56525 1 This...

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Unformatted text preview: choi (kc24547) – HW14 – BERG – (56525) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine whether the series ∞ summationdisplay n =2 ( − 1) n n 8 ln n is conditionally convergent, absolutely con- vergent, or divergent. 1. series is absolutely convergent 2. series is conditionally convergent 3. series is divergent correct Explanation: By the Divergence Test, a series ∞ summationdisplay n = N ( − 1) n a n will be divergent for each fixed choice of N if lim n →∞ a n negationslash = 0 since it is only the behaviour of a n as n → ∞ that’s important. Now, for the given series, N = 2 and a n = n 8 ln n . But by L’Hospital’s Rule, lim x →∞ x ln x = lim x →∞ 1 1 /x = ∞ . Consequently, by the Divergence Test, the given series is divergent . 002 10.0 points Determine whether the series ∞ summationdisplay n =0 3 √ n + 1 cos nπ is conditionally convergent, absolutely con- vergent or divergent. 1. divergent 2. absolutely convergent 3. conditionally convergent correct Explanation: Since cos nπ = ( − 1) n , the given series can be rewritten as the alternating series ∞ summationdisplay n =0 ( − 1) n 3 √ n + 1 = ∞ summationdisplay n =0 ( − 1) n f ( n ) with f ( x ) = 3 √ x + 1 . Now f ( n ) = 3 √ n + 1 > 3 √ n + 2 = f ( n + 1) for all n , while lim n →∞ f ( n ) = lim n →∞ 3 √ n + 1 = 0 . Consequently, by the Alternating Series Test, the given series converges. On the other hand, by the Limit Comparison Test and the p-series test with p = 1 / 2, we see that the series ∞ summationdisplay n =0 f ( n ) is divergent. Consequently, the given series is conditionally convergent . 003 10.0 points To apply the root test to an infinite series ∑ n a n the value of ρ = lim n →∞ ( a n ) 1 /n choi (kc24547) – HW14 – BERG – (56525) 2 has to be determined. Compute the value of ρ for the series ∞ summationdisplay n =1 3 n + 2 n parenleftbigg 3 2 parenrightbigg n . 1. ρ = 4 3 2. ρ = 3 2 correct 3. ρ = 2 3 4. ρ = 3 5. ρ = 9 2 Explanation: After division, 3 n + 2 n = 3 parenleftbigg 1 + 2 3 n parenrightbigg , so ( a n ) 1 /n = parenleftbigg 3 braceleftbigg 1 + 2 3 n bracerightbiggparenrightbigg 1 /n 3 2 . But lim n →∞ 3 1 /n parenleftbigg 1 + 2 3 n parenrightbigg 1 /n = 1 as n → ∞ . Consequently, ρ = 3 2 . 004 10.0 points To apply the ratio test to the infinite series summationdisplay n a n , the value λ = lim n →∞ a n +1 a n has to be determined. Compute λ for the series ∞ summationdisplay n =1 2 n 6 n 2 + 7 . 1. λ = 0 2. λ = 2 7 3. λ = 2 13 4. λ = 1 3 5. λ = 2 correct Explanation: By algebra, a n +1 a n = 2 n +1 2 n bracketleftbigg 6 n 2 + 7 6( n + 1) 2 + 7 bracketrightbigg ....
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cal14 - choi(kc24547 – HW14 – BERG –(56525 1 This...

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