15_new - HW03 gilbert (55035) This print-out should have 18...

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HW03 – gilbert – (55035) 1 This print-out should have 18 questions. (ii) while if Multiple-choice questions may continue on X k before thenext answering. column or page find all choices 0 < c k b k , c k converges, then the Comparison Test is inconclusive be- 001 10.0 points cause X b k could converge, but it could di- If a k , b k , and c k satisfy the inequalities verge - we can’t say precisely without further restrictions on b k . 0 < a k c k b k , Consequently, what we can say is for all k, what can we say about the series (A) converges , (B) need not converge . (A) : k X =1 a k , (B) : k X =1 b k 002 10.0 points Determine whether the series if we know that the series 2 n X =1 3 n(n + 1)(n + 5) (C) : k X =1 c k p converges or diverges. is convergent but know nothing else about a k 1. series is convergent and b k ? 1. (A) diverges , (B) converges 2. series is divergent correct Explanation: Note first that 2. (A) need not converge , (B) converges n 3 lim = 1 > 0 . 3. (A) converges , (B) need not converge n → ∞ n(n + 1)(n + 5) correct Thus by the limit comparison test, the given series 2 4. (A) converges , (B) converges n X =∞ p 5. (A) converges , (B) diverges converges if and only if the series 2 6. (A) diverges , (B) diverges n X =1 n Explanation: converges. But by the p-series test with p = 1 Let’s try applying the Comparison Test: (or use the comparison test applied to the har- (i) if monic series), this last series diverges. Conse- X k quently, the given 0 < a k c k , c k converges, series is divergent . then the Comparison Test applies and says that X a k converges; 003 10.0 points
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2 Determine whether the series X k n X 3 (C) 4 n = 15 k = 1 (k + 2)3 k converge(s)? converges or diverges. 1. series is divergent 2. series is convergent correct Explanation: We use the Limit Comparison Test with k 1 a k = , b k =. (k + 2)3 k 3 k For lim a k = lim k = 1 > 0 . k → ∞ b k k →∞ k + 2 Thus the series k X =1 (k + k 2)3 k converges if and only if the series X 1 k = 1 3 k converges. But this last series is a geometric series with |r| = 1 3 < 1 , hence convergent. Consequently, the given series is series is convergent . 004 10.0 points Which of the following series (A) n X =1 3n 2 2 n + 6 (B) 5 2 n n X =1 1. A and B only 2. A, B, and C 3. B and C only correct 4. C only 5. B only Explanation: (A) Because of the way the n TH term is defined as a quotient of polynomials in the series, use of the integral test is suggested. Set 2x f (x) = 3x 2 + 6 . Then f is continuous, positive and decreasing on [1, ∞); thus X 2n n = 1 3n 2 + 6 converges if and only if the improper integral 2 x 1 3x 2 + 6 dx converges, which requires us to evaluate the integral n 2 x I n = 1 3x 2 + 6 dx . Now after substitution (set u = x
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15_new - HW03 gilbert (55035) This print-out should have 18...

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