MATH 201 Exam 2, Spring 2016

— ee f Ba — Spring 2015-2016 Quis + oe aan den Prof, M. Egeileh Math 201 - Sections 5 to 8 Problem 1 (answer on pages 1 and 2 of the booklet) Which of the following series converge, cai ie diverge? When possible find the are of the series. (8 pts each) ‘, ( phyA x ue a8 e oo cata P. ede. ee — (-1y* 8 ' cass 0.5 1+—— A)S~ ( Sa eps i >> ) a) ye In(1+—5 n=2 n=] q n=2 Problem 2 (answer on pages 3 and 4 of the booklet) i. Vi) n (ex —1) (x + 4)”. (20 pts) co Wind the interval of convergence of the power series : y ( n=3 Problem 3 (answer on pages 5 and 6 of the booklet ) & pant) He uae {CA \ 1. Prove that Arctana = ) —1)" — <d 9 pts) " An © ( ) 5 T? |x| ( Pp n=0 2, Find the Taylor polynomial p;(x) generated by f(x) = Arctan 2 at the point x = 0. Then N4 use the alternating series estimation theorem to estimate the error resulting form the ap- proximation Arctan(—0.1) © p:(—0.1). Is p:(—0.1) an overestimate or an underestimate / fi (7 pts) 3. Use Taylor’s theorem to estimate the error resulting from the approximation Arctan(1/V3) pi(1/V3). Is pi(1/ V3) an overestimate or an underestimate ? (Notice that pi(x) = po(x). You may need the fact that the third derivative of the function f(x) = Arctan x is f(x) = (6x* + 4a” = 2)/(1+ 2*)*). (6 pts) : i 5 Arctan £ 4, Find a power series expansion for the function g(x) = sf Ye mer) about the point xz = 0. It is enough to, find the first three terms of the power series expansion. (6 pts ) Problem 4 (answer on last page of the booklet) : a “Show that for n > 1, we have the following inequality : In(1) + In(2) + +++ + nals —1)+In(r) § min(n) —n+1+In(n). (5 pts) 2. Does the series ee n=1 ae converge ? Why or why not? (5 pts) x Problem 5 (answer on back of last page of the booklet ) Which of the following sequences converge, and which diverge ? Find sequence. (7 pts each) the limit of each convergent a) {an}, given in list notation by 0, 2,0, 2,0, 270) aeons pts) ve a iby 2.) b) {sn}, given in list notation by 5,7) 31g 39° 64” Jog? rth a pis)