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Real Final Exam Solutions

Real Final Exam Solutions - MATH 31B SECTION 2 FINAL EXAM...

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MATH 31B SECTION 2 FINAL EXAM SOLUTIONS. Please note: Show your work. Correct answers not accompanied by sufficent explanations will receive little or no credit. Please call one of the proctors if you have any questions about a problem. No calculators, computers, PDAs, cell phones, or other devices will be permitted. #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 Total SID: TA: Section(circle): Tuesday Thursday Name: 1
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MATH 31B SECTION 2 FINAL EXAM SOLUTIONS. 2 Problem 1. Let f ( x ) = X n =0 3 n 2 n + 5 x n . Find the value of f 000 (0) . Solution. If c n denotes the coefficient of x n in this power series, then f 000 (0) = 3! · c 3 = 3! · 3 · 3 2 · 3+5 = 54 11 .
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MATH 31B SECTION 2 FINAL EXAM SOLUTIONS. 3 Problem 2. Determine if the series X n =0 1 ( n + 1)( n + 2) is convergent or divergent. If it is convergent, find its sum. Solution. We have that 1 ( n + 1)( n + 2) = 1 n + 1 - 1 n + 2 . The series is telescoping: 1 1 - 1 2 + 1 2 - 1 3 + 1 3 - 1 4 + · · · + 1 n + 1 - 1 n + 2 = 1 - 1 n + 2 1 . Hence the sum of the series is 1 .
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MATH 31B SECTION 2 FINAL EXAM SOLUTIONS. 4 Problem 3. Let f ( x ) = tan - 1 x . Find a power series representation for f around 0 . ( Hint: represent tan - 1 x as an integral). Solution. We have: tan - 1 x = Z 1 1 + x 2 dx + C = Z 1 1 - ( - x 2 ) dx + C = Z X n =0 ( - 1) n x 2 n + C = X n =0 ( - 1) n x 2 n +1 2 n + 1 + C. To find C , we evaluate both sides at x = 0 ; since tan - 1 0 = 0 , C = 0 and tan - 1 x = X n =0 ( - 1) n x 2 n +1 2 n + 1 .
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MATH 31B SECTION 2 FINAL EXAM SOLUTIONS. 5 Problem 4. Is the improper integral Z 0 e sin x dx convergent or divergent? Explain. Hint: sin x is periodic.
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