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cal12 - choi(kc24547 HW12 BERG(56525 This print-out should...

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choi (kc24547) – HW12 – BERG – (56525) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 3) 10.0 points Write each of the following finite sums in summation notation. (i) The sum of the first ten positive odd inte- gers. 1. sum = 10 summationdisplay i =1 i 2. sum = 10 summationdisplay i =1 (2 i + 1) 3. sum = 10 summationdisplay i =1 (2 i 1) correct 4. sum = 10 summationdisplay i =1 2 i 5. sum = 10 summationdisplay i =1 ( i 1) Explanation: Positive odd integers are represented by 2 i 1, i = 1 , 2 , 3 , . . . . Thus the required sum is given by sum = 10 summationdisplay i =1 (2 i 1). 002 (part 2 of 3) 10.0 points (ii) The sum of the cubes of the first n positive integers. 1. None of these 2. sum = n summationdisplay i =0 i 3 3. sum = n summationdisplay i =1 i 3 correct 4. sum = n summationdisplay i =1 i 5. sum = n summationdisplay i =0 i Explanation: The positive integers are represented by i , i = 1 , 2 , 3 , . . . . Thus the required sum is given by sum = n summationdisplay i =1 i 3 . 003 (part 3 of 3) 10.0 points (iii) 6 + 10 + 14 + 18 + . . . + 42 . 1. sum = 10 summationdisplay i =1 4 i 2. sum = 10 summationdisplay i =1 (6 + 4 i ) 3. sum = 10 summationdisplay i =1 (2 + 4 i ) correct 4. None of these 5. sum = 10 summationdisplay i =1 6 i Explanation: The difference between consecutive terms is 4, so 6+10 + 14 + 18 + · · · + 42 = (2 + 4) + (2 + 8) + (2 + 12)+ . . . + (2 + 40) = sum = 10 summationdisplay i =1 (2 + 4 i ).
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choi (kc24547) – HW12 – BERG – (56525) 2 Alternate Solution: 6+10 + 14 + 18 + · · · + 42 = (6 + 0) + (6 + 4) + (6 + 8)+ . . . + (6 + 36) = 10 summationdisplay i =1 [6 + 4 ( i 1)] = sum = 10 summationdisplay i =1 (2 + 4 i ). 004 10.0 points Determine whether the infinite series 4 2 + 1 1 2 + 1 4 · · · is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 8 3 correct 2. convergent with sum 3 8 3. divergent 4. convergent with sum 1 8 5. convergent with sum 8 Explanation: The infinite series 4 2 + 1 1 2 + 1 4 · · · = summationdisplay n =1 a r n - 1 is an infinite geometric series with a = 4 and r = 1 2 . But a geometric series summationdisplay n =1 a r n - 1 (i) converges with sum a 1 r when | r | < 1, (ii) and diverges when | r | ≥ 1 . Consequently, the given series is convergent with sum 8 3 . 005 10.0 points Determine whether the series summationdisplay n =0 4 (cos ) parenleftbigg 3 4 parenrightbigg n is convergent or divergent, and if convergent, find its sum. 1. convergent with sum 7 16 2. divergent 3. convergent with sum 16 7 correct 4. convergent with sum 16 5. convergent with sum 16 7 6. convergent with sum 16 Explanation: Since cos = ( 1) n , the given series can be rewritten as an infinite geometric series summationdisplay n =0 4 parenleftbigg 3 4 parenrightbigg n = summationdisplay n =0 a r n in which a = 4 , r = 3 4 . But the series n =0 ar n is (i) convergent with sum a 1 r when | r | < 1,
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choi (kc24547) – HW12 – BERG – (56525) 3 and (ii) divergent when | r | ≥ 1.
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