MATH 201 Exam 1, Spring 2020

Math 201—Exam 1 (Spring 19) Profs. R. Aoun, B. Shayya, and T. Tlas e Please write your section number on your booklet. e Please place your student ID card on the desk in front of you. e Please answer each problem on the indicated page(s) of the booklet. Any part of your answer not written on the indicated page(s) will not be graded. e Unjustified answers will receive little or no credit. Problem 1 (answer on pages 1 and 2 of the booklet.) (8 pts each) Which of the following sequences converge, and which diverge? Find the limit of each convergent sequence. (i) a, =(n 41)" (ii) b, = (iii) c= (1 +8in =)" Problem 2 (answer on pages 3, 4, and 5 of the booklet.) (8 pts each) Which of the following series converge, and which diverge? When possible, find the sum of the series. (a) Date (GE + $2") 0) Sess 1 (2")(m1)(nt) (c) =, (Ps (d) es (si it givtn+3)) Problem 3 (answer'on pages 6 and 7 of the booklet.) Consider the power series al n F(z) = =e n=1 (i) (16 pts) Find the radius and interval of convergence of F(x). {ii) (8 pts) Use the alternating series estimation theorem {(ASET) to estimate F(2) with an er- ror of magnitude less than 0.25. Is your answer an over-estimate or an under-estimate? (Hint: In4 = 21n2.) Problem 4 (answer on pages 8 and 9 of tle booklet.) (i) (12 pts) Use the integral test to prove that the p-series Pai hy By op Le ga Spt gp diverges if 0 < p < 1, and converges if p > 1. (ii) (8 pts) Prove that =e! pk (>: 5) =k