Prof. Ayoub / MAT1575 Calculus II / NYCCT Name: ____________________ Test#3/ April 20th , 2016 Due Monday May 02nd, 2016 Any late submission will reduce the test score by 50%. You must show all necessary work for full credit. Staple additional worksheets if needed. All integrals must be calculated manually. Correct answer with no work will receive only 25% credit. Max. 100 points 1. (6 pts.)Evaluate the indefinite integral: x e 2. (6 pts.)Evaluate the indefinite integral: x 2 7x 2 dx 7x 4 dx 11x 28 3. (6 pts.)Show that the series converge 4. (6 pts.)Determine if the improper integral converges, if so, evaluate it. 6 1 3 x8 5. Determine if the series converge or diverge. Justify your answer using an appropriate test; 3n 7 a. n 1 6n 3 n (6 pts.) n3 6n b. n! n 1 (6 pts.) 6. Determine if the series converges absolutely, conditionally, or not at all. n 1 1n 8 n6 (6 pts.) 7. Find the radius and interval of convergence of the following series; xn n n 1 2 (6 pts.) 8. Apply the ratio test to determine the convergence or divergence, or state that the ratio test is inconclusive . n!3 n 1 3n ! (6 pts.) 9. Find the Taylor polynomial of degree 3 for the function f ( x) sin(3x) centered at a hint: Tn ( x) f (a) n f ' (a) f ( x) ( x a)1 ... ( x a) n 1! n! (6 pts.) x2n 10. Use the ratio test to show that n has a radius of convergence R 3 n 0 3 11. Show that x 3n 1 64 n0 n has a radius of convergence R 4 (6 pts.) (6 pts.) 12. For which value(s) of x does n! x n 0 n converge? (6 pts.) 9 13. Use the ratio test to show that 2 1 xn n n has a radius of convergence R 2. Then determine whether it converges at R 2. (10 pts.) 14. Find the interval of Convergence for the following series: (6 pts.) (6 pts.)