University of California Los Angeles Winter 2025 Instructor Arant Johnson Date March 6 2025 Name UCLA ID Signature MATH 31B Integration and Infinite Series Midterm 2 This exam contains 10 pages including this cover page and 4 problems Check to see if any pages are missing Enter all requested information on the top of this page and also Put your initials on the top of every page in case the pages become separated This is a closed book exam No cheat sheets calculators phones laptops etc are allowed The following rules apply Problem Points Score Unless the problem explicitly states otherwise you must show your work to justify your answers Mysterious or unsupported answers will not re ceive full credit An incorrect answer supported by substantially correct calculations and explana tions might still receive partial credit Organize your work in a reasonably neat and coherent way in the space provided Work scat tered all over the page without a clear ordering will receive very little credit If you need more space you may use the back of the pages clearly indicate when you have done this There is also a black page on the back of this exam that you may use for scratch paper Good luck 1 2 3 4 Total 9 10 24 12 55 MATH 31B Midterm 2 Page 2 of 10 March 6 2025 Double angle formulas Integrals sin2 1 2 1 cos 2 sin 2 2 cos sin n 1 n sinn 2 x dx n cosn 2 x dx n 1 n n dx dx 1 2 1 2 x2 1 tanm 1 x sin x cos x C sin x cos x C sin 1 x C tan 1 x C x 2 cosn 1 x sin x x 2 sinn 1 x cos x 1 x2 sin2 x dx sinn x dx cos2 x dx cosn x dx tan x dx ln sec x C tanm x dx cot x dx ln sin x C cotm x dx sec x dx ln sec x tan x C secm x dx csc x dx ln csc x cot x C cscm x dx sin mx sin nx dx sin mx cos nx dx cos mx cos nx dx sin m n x 2 m n cos m n x 2 m n 2 m n sin m n x tan x secm 2 x cot x cscm 2 x m 1 m 1 m 1 tanm 2 x dx m 1 cotm 2 x dx cotm 1 x m 2 m 1 secm 2 x dx m 1 cscm 2 x dx m 2 sin m n x 2 m n cos m n x 2 m n sin m n x 2 m n C m n C m n C m n MATH 31B Midterm 2 Page 3 of 10 March 6 2025 1 Fill in the requested information No justi cation or work is required a 1 point 1 np converges when n 1 b 2 points The sum of the in nite series n 1 2 3 n is c 3 points Fully simplify The rst 3 partial sums of the series n 1 2 n 2 n 1 are S1 S2 S3 d 3 points The rst three terms of the sequence 10 n 7n n 1 are MATH 31B Midterm 2 Page 4 of 10 March 6 2025 2 10 points Consider the series 1 n 1n1 3 n4 3 n n 1 Does this series converge absolutely converge conditionally or diverge Justify your answer completely You do not need to calculate any in nite sums MATH 31B Midterm 2 Page 5 of 10 March 6 2025 3 For each series below determine if it converges or diverges Specify any and all convergence tests you use and show all of your work and justi cation You do not need to calculate any in nite sums a 8 points 5n 3n n n 1 Answer for a b 8 points cos n n2 n 1 Answer for b MATH 31B Midterm 2 Page 6 of 10 March 6 2025 c 8 points 1 k ln k2 k 2 Answer for c MATH 31B Midterm 2 Page 7 of 10 March 6 2025 4 Determine if the improper integral is convergent If it is convergent compute its value Show all your work and justi cation a 6 points 2 0 5 x 2 dx b 6 points 1 e 2xdx Answer for a Answer for b MATH 31B Midterm 2 Page 8 of 10 March 6 2025 This page intentionally left blank MATH 31B Midterm 2 Page 9 of 10 March 6 2025 This page intentionally left blank MATH 31B Midterm 2 Page 10 of 10 March 6 2025 This page intentionally left blank