M408L hwk11 solutions

M408L hwk11 solutions - Pham, Quoc – Homework 11 – Due:...

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Unformatted text preview: Pham, Quoc – Homework 11 – Due: Apr 10 2007, 3:00 am – Inst: Eric Katerman 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 3 3 + 5- 4 4 + 5 + 5 5 + 5- 6 6 + 5 + ... + (- 1) 5 8 8 + 5 using summation notation. 1. 5 X k = 0 (- 1) k- 3 k k + 5 2. 5 X k = 0 k k + 5 3. 9 X k = 4 (- 1) k- 3 k k + 5 4. 8 X k = 3 k k + 5 5. 8 X k = 3 (- 1) k- 3 k k + 5 correct 6. 8 X k = 3 (- 1) k- 4 k k + 5 Explanation: The numerators form a sequence 3 , 4 , 5 , 6 , ... , 8 , while the denominators form a sequence 3 + 5 , 4 + 5 , 5 + 5 , 6 + 5 , ... , 8 + 5 . Thus the general term in the series is of the form a k = (- 1) k- 3 k k + 5 where the sum ranges from k = 3 to k = 8. Consequently, the series becomes 8 X k = 3 (- 1) k- 3 k k + 5 in summation notation. keywords: finite sum, summation notation, 002 (part 1 of 1) 10 points Find the sum of the finite series 3 + 3 · 7 9 + 3 · 7 2 9 2 + ... + 3 · 7 9 9 9 . 1. sum = 3 9 9 ‡ 9 9- 7 9 2 · 2. sum = 3 9 9 ‡ 9 8- 7 8 2 · 3. sum = 3 ‡ 9 10- 7 10 2 · 4. sum = 3 ‡ 9 9- 7 9 2 · 5. sum = 3 9 9 ‡ 9 10- 7 10 2 · correct Explanation: The given series is a finite geometric series 9 X n = 0 ar n , with a = 3 , r = 7 9 . Now 9 X n = 0 ar n = a ‡ 1- r 10 1- r · . Consequently, sum = 3 9 9 ‡ 9 10- 7 10 2 · . keywords: finite geometric series Pham, Quoc – Homework 11 – Due: Apr 10 2007, 3:00 am – Inst: Eric Katerman 2 003 (part 1 of 3) 10 points If the n th partial sum of ∑ ∞ n = 1 a n is S n = 5 n- 3 n + 1 , (i) what is a 1 ? 1. a 1 =- 1 2. a 1 =- 4 3. a 1 = 5 4. a 1 = 1 correct 5. a 1 = 4 Explanation: Since a 1 = S 1 , a 1 = 1 . 004 (part 2 of 3) 10 points (ii) What is a n for n > 1? 1. a n = 2 n 2 2. a n = 8 n ( n- 1) 3. a n = 2 n ( n + 1) 4. a n = 2 n ( n- 1) 5. a n = 8 n ( n + 1) correct 6. a n = 8 n 2 Explanation: Since S n = a 1 + a 2 + ··· + a n , we see that a n = S n- S n- 1 . But S n = 5 n- 3 n + 1 = 5( n + 1)- 8 n + 1 = 5- 8 n + 1 . Consequently, a n = 8 n- 8 n + 1 = 8 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 = 6 2. sum = 8 3. sum = 4 4. sum = 7 5. sum = 5 correct Explanation: By definition sum = lim n →∞ S n = lim n →∞ ‡ 5 n- 3 n + 1 · . Thus sum = 5 . keywords: 006 (part 1 of 1) 10 points Determine whether the series ∞ X n = 0 2 (cos nπ ) µ 3 4 ¶ n is convergent or divergent, and if convergent, find its sum. 1. divergent Pham, Quoc – Homework 11 – Due: Apr 10 2007, 3:00 am – Inst: Eric Katerman 3 2. convergent with sum- 8 7 3. convergent with sum- 8 4. convergent with sum 8 5. convergent with sum 8 7 correct 6. convergent with sum- 7 8 Explanation: Since cos nπ = (- 1) n , the given series can be rewritten as an infinite geometric series ∞ X n =0 2 µ- 3 4 ¶ n = ∞ X n = 0 ar n in which a = 2 , r =- 3 4 ....
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This homework help was uploaded on 03/19/2008 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

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M408L hwk11 solutions - Pham, Quoc – Homework 11 – Due:...

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