Math 2300: Final Exam - Integration, Differential Equations, Series and Polar Coordinates , Exams of Analytical Geometry and Calculus

Download Math 2300: Final Exam - Integration, Differential Equations, Series and Polar Coordinates and more Exams Analytical Geometry and Calculus in PDF only on Docsity! Math 2300 page 1 of 9 1. (20) Evaluate the following integrals: (i) ∫ sin2 θ cos3 θ dθ; (ii) ∫ e 0 lnx dx; Math 2300 page 2 of 9 (iii) ∫ ∞ 0 4 x2 + 16 dx; (iv) ∫ sin(lnx) dx. Math 2300 page 5 of 9 4. (15) (i) Does the sequence { n sin ( πn 2 ) 2n2 + 1 } ∞ n=1 converge or diverge? (ii) Does the series ∞ ∑ k=0 (−1)k (2k + 1)! (π 4 )2k+1 converge or diverge? If it converges, find its sum. Math 2300 page 6 of 9 5. (10) Determine if the following series diverge, converge conditionally, or converge ab- solutely. (i) ∞ ∑ k=1 tan−1 k k2 ; (ii) ∞ ∑ k=1 (−1)k(k2 + 1) 2k2 + k − 1 . Math 2300 page 7 of 9 6. (10) Find the interval of convergence for ∞ ∑ k=2 (−1)k k (2x + 3)k. 7. (20) (i) Evaluate the definite integral ∫ π/2 0 x cos x dx using integration by parts.