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 cos 2 1 2 sin 2 2 cos sin n n 1 2 x 2 cid 90 cid 90 x2 1 1 2 sin x cos x C n 1 n cos2 x dx sin x cos x C sin 1 x C tan 1 x C dx 1 x2 dx x 2 cosn 1 x sin x cosn x dx tan x dx ln sec x C sin2 x dx sinn x dx sinn 1 x cos x cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 tanm x dx cid 90 cot x dx ln sin x C cotm x dx cotm 1 x cid 90 m 1 cid 90 sec x dx ln sec x tan x C cid 90 secm x dx cid 90 csc x dx ln csc x cot x C cscm x dx cot x cscm 2 x sin m n x sin mx cos nx dx cos m n x 2 m n sin m n x tanm 1 x m 1 tan x secm 2 x sin mx sin nx dx 2 m n m 1 m 1 cid 90 cos mx cos nx dx 2 m n cid 90 cid 90 cid 90 cid 90 n 1 n cid 90 sinn 2 x dx cosn 2 x dx tanm 2 x dx cotm 2 x dx cid 90 cid 90 m 2 m 1 m 2 m 1 secm 2 x dx cscm 2 x dx sin m n x C m n 2 m n cos m n x C m n 2 m n sin m n x 2 m n C m n MATH 31B Midterm 2 Page 3 of 10 March 6 2025 1 Fill in the requested information No justification or work is required cid 88 n 1 a 1 point 1 np converges when b 2 points The sum of the infinite series cid 18 2 cid 19 n 3 cid 88 n 1 is c 3 points Fully simplify The first 3 partial sums of the series cid 18 2 cid 88 n 1 cid 19 2 n n 1 are S1 S2 S3 d 3 points The first three terms of the sequence 10 are cid 26 cid 27 n 7n n 1 MATH 31B Midterm 2 Page 4 of 10 March 6 2025 2 10 points Consider the series cid 88 n 1 1 n 1n1 3 n4 3 n Does this series converge absolutely converge conditionally or diverge Justify your answer completely You do not need to calculate any infinite 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 justification You do not need to calculate any infinite sums a 8 points cid 88 n 1 5n 3n n b 8 points cid 88 n 1 cos n n2 Answer for a Answer for b MATH 31B Midterm 2 Page 6 of 10 March 6 2025 c 8 points cid 88 k 2 1 k ln k2 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 justification a 6 points cid 90 2 0 5 x 2 dx b 6 points e 2xdx cid 90 1 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