HW 11 key - Miller, Kierste – Homework 11 – Due: Apr 10...

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Unformatted text preview: Miller, Kierste – Homework 11 – Due: Apr 10 2007, 3:00 am – Inst: Gary Berg 1 This print-out should have 24 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Rewrite the finite sum 15 3 + 3- 20 4 + 3 + 25 5 + 3- 30 6 + 3 + ... + (- 1) 5 40 8 + 3 using summation notation. 1. 5 X k = 0 (- 1) k- 3 5 k k + 3 2. 8 X k = 3 5 k k + 3 3. 8 X k = 3 (- 1) k- 4 5 k k + 3 4. 9 X k = 4 (- 1) k- 3 5 k k + 3 5. 5 X k = 0 5 k k + 3 6. 8 X k = 3 (- 1) k- 3 5 k k + 3 correct Explanation: The numerators form a sequence 15 , 20 , 25 , 30 , ... , 40 , while the denominators form a sequence 3 + 3 , 4 + 3 , 5 + 3 , 6 + 3 , ... , 8 + 3 . Thus the general term in the series is of the form a k = (- 1) k- 3 5 k k + 3 where the sum ranges from k = 3 to k = 8. Consequently, the series becomes 8 X k = 3 (- 1) k- 3 5 k k + 3 in summation notation. keywords: finite sum, summation notation, 002 (part 1 of 1) 10 points Find the sum of the finite series 8 + 8 · 5 8 + 8 · 5 2 8 2 + ... + 8 · 5 8 8 8 . 1. sum = 8 8 8 ‡ 8 7- 5 7 3 · 2. sum = 8 8 8 ‡ 8 8- 5 8 3 · 3. sum = 8 8 8 ‡ 8 9- 5 9 3 · correct 4. sum = 8 ‡ 8 9- 5 9 3 · 5. sum = 8 ‡ 8 8- 5 8 3 · Explanation: The given series is a finite geometric series 8 X n = 0 ar n , with a = 8 , r = 5 8 . Now 8 X n = 0 ar n = a ‡ 1- r 9 1- r · . Consequently, sum = 8 8 8 ‡ 8 9- 5 9 3 · . keywords: finite geometric series Miller, Kierste – Homework 11 – Due: Apr 10 2007, 3:00 am – Inst: Gary Berg 2 003 (part 1 of 3) 10 points If the n th partial sum of ∑ ∞ n = 1 a n is S n = 2 n- 5 n + 1 , (i) what is a 1 ? 1. a 1 = 7 2 2. a 1 =- 3 2 correct 3. a 1 = 3 2 4. a 1 = 2 5. a 1 =- 7 2 Explanation: Since a 1 = S 1 , a 1 =- 3 2 . 004 (part 2 of 3) 10 points (ii) What is a n for n > 1? 1. a n = 3 n 2 2. a n = 7 n 2 3. a n = 3 n ( n- 1) 4. a n = 7 n ( n + 1) correct 5. a n = 7 n ( n- 1) 6. a n = 3 n ( n + 1) Explanation: Since S n = a 1 + a 2 + ··· + a n , we see that a n = S n- S n- 1 . But S n = 2 n- 5 n + 1 = 2( n + 1)- 7 n + 1 = 2- 7 n + 1 . Consequently, a n = 7 n- 7 n + 1 = 7 n ( n + 1) for all n > 1. 005 (part 3 of 3) 10 points (iii) What is the sum ∑ ∞ n = 1 a n ? 1. sum =- 1 2. sum = 1 3. sum = 2 correct 4. sum = 0 5. sum =- 2 Explanation: By definition sum = lim n →∞ S n = lim n →∞ ‡ 2 n- 5 n + 1 · . Thus sum = 2 . keywords: 006 (part 1 of 1) 10 points Determine whether the series ∞ X n = 0 4 (cos nπ ) µ 1 2 ¶ n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 8 Miller, Kierste – Homework 11 – Due: Apr 10 2007, 3:00 am – Inst: Gary Berg 3 2. divergent 3. convergent with sum- 3 8 4. convergent with sum 8 3 correct 5. convergent with sum- 8 3 6. convergent with sum- 8 Explanation: Since cos nπ = (- 1) n , the given series can be rewritten as an infinite geometric series ∞ X n =0 4 µ- 1 2 ¶ n = ∞ X n = 0 ar n in which...
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This note was uploaded on 05/08/2008 for the course M 408 L taught by Professor Cepparo during the Spring '08 term at University of Texas.

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HW 11 key - Miller, Kierste – Homework 11 – Due: Apr 10...

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