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practice_prelim3_fa05

# practice_prelim3_fa05 - 1 2 n 1 Â[10 points converge Why or...

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Practice Prelim 3, Math 191, Fall 2005 No calculators. Clearly mark each answer. 1. Decide, giving reasons, whether the following series converges abso- lutely, converges conditionally, or diverges? a ) X n =1 (ln n ) 2 n 3 / 2 [10 points ] b ) X n =2 1 n + sin n [5 points ] c ) X n =2 n n + 1 n 3 + 3 n + 1 [5 points ] d ) X n =1 (2 n + 1)! 2 (3 n )! [10 points ] 2. a) Find the Maclaurin series for the function f ( x ) = x 2 1 + x For what values of x does the series converge absolutely? b) Does the series converge at x = 1? Explain. [20 points] 3. a) Find the sum of the series 1 + 2 10 + 3 10 2 + 2 10 4 + 3 10 5 + 2 10 7 + 3 10 8 + ... [10 points ] b) Does the series b ) X n =1 sin 1 2 n - sin

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Unformatted text preview: 1 2 n + 1 Â¶ [10 points ] converge? Why or why not? 4. Evaluate the integral-Z 1 ln xdx [10 points ] 1 5. According to the error-bound formula for Simpsonâ€™s rule (the formula would be given on the actual exam), how many sub-intervals should you use to be sure of estimating the value of ln3 = Z 3 1 1 x dx by Simpsonâ€™s rule with an error no more than 10-2 in absolute value? (Remember that for Simpsonâ€™s rule, the number of sub-intervals has to be even). [20 points] 2...
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