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Reshetikhin07Final

Reshetikhin07Final - Math 1B Final Examination...

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Math 1B, Final Examination N.Reshetikhin, December 13, 2007 Problem 1 2 3 4 5 6 7 8 9 10 11 12 Total Points 15 20 25 20 20 20 20 20 20 20 20 20 100 Grade Student’s Name: GSI’s name: Student’s i.d. number: 1. 10 pnts Evaluate the integral Z 2 x 3 sin( x 2 ) dx 2. 15 pnts Evaluate the integral Z 1 ( t 2 + 1)( t - 1) dx 3. 15 pnts Indicate which of the following statements are true and which are false. Do not show your work. 1. Z 1 1 + sin( 1 x ) x 3 dx converges by comparison test with Z 1 1 x 3 dx . 2. Z 1 0 sin( x ) x dx diverges by comparison test with Z 1 0 1 x dx . 3. Z 1 0 dx x 2 + x dx is a convergent improper integral. 4. Z 1 1 ( x - 1) 2 dx is a divergent improper integral. 5. Z 1 0 ln( x ) x 1 / 2 dx is a convergent improper integral. 1
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4. 15 pnts Find the radius and the interval of convergence of the power series X n =1 n - 1 ( n + 2)(2 n + 5) ( x 2 ) n 5. 15 pnts State whether the following series is absolutely convergent, condi- tionally convergent, or divergent. Do not show your work.
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