MATH 201 Exam 1 With Solution, Fall 2015

Math 201-Fall-2015-16 Calculus and Analytic Geometry III, sections 25,26,27,28 Quiz 1, October 1, 2015-Duration: 60 minutes YOUR NAME: K wy / YOUR AUB ID #: INST : — Closed book and notes. NO CACULATORS ALLOWED. Turn OFF and put away any cell GRADES: 123 4/5 | 1. ( 20 pts.) Find the following limits (a) ie 2n— | lun set) 14S ou — noo Ar— | (b) fy gece eels gs less sage linn —2 Sa Si nel n tio < tphaher gt [Le Jz tog tig Ft te = a a J n+! n az | < Z 14 2x | arn 6vn . vn Tike Lins “rn a~e , He matt 3 at J / 2. (a) (15 pts.) Test each of the following series for convergence oo) yet +1 Sab j# By vaio gat : A ea - din 4. Carl + — bind. nxeett Eircs r Ite a oJ = Z ney} andl 2 1 Atv Wrnvnnger , 24 Integral lot : [tn = bon. | ae + - dum (Lndnx]” ) - lin [tntnh - tint 2/ =a Arve a Ave emer a Hye - I, Mode, 1 uy Ukirtrrng peowue “tnx o pn Lawn ge Xbnp (c) ( 10 pts.) Find all values of p for which the following series converges ~ P | Hi cull (eye) n We ug L.C.T oe oe ne wrth yw? rel OR goolk * — iL f ae: a dane [2-18 7G AT" \ dar. $ fre ) ye x? xe x chy? Ly". rf - |, ; But 2 vomege tbh Zp>!, of P7#H - 4 4 Root test, pis 3: Igfis,) Find all values of x for which the following series is convergent. 4"(n!)?2( — 1)" = Ga! gles + ew Hy Ue mbro leat + dun |e or 0-0" [AG Oe0 | Pr n>e0 (272), (20)! wall 2 ca (6 = haw I(x—1) ) ; 4. (0H) <= 4)x-tle jx na oe (204) (2742) 4 Ane oriuser abs obalily, “ps Ach % ula gix- (<(, Jx-tleg yp tox (</ 0<xe2 .(© End pews - oH} KoR + > 4h now (An)! ; , 7 . ») _ 4Gnu'). Atal). Arp! we wan Wat ( inpr)! . er) (tn py) (2H 40) = ony Inet >! 4n'L nt r On lr mine Wi vdute-- The tom 9 Be N/E lien 0 x64 jew, the ame MVOGZ- 4 KO, = path 4° iy (2)! BL drm the ore more m dsechele rte 20 © lum Gee ont lind » gre, Abner iaze Ken One tae AGS Ads chabihy Vs% SCS k eR inf othyrwre A verge, 4 y» w? 5 4. (a) (AB pts.) Find the Taylor series expansion of f(x) = and use it to find ((3). ( Hint: First find the Taylor series of —L about a= 3): 1 2.ti*. 1+ ee MHS aid MAS? 3 ne = § Row Aid vege [x -3)e Y- mp di) < (x3 - (-1)” (x- 3) 2 GL as ger! By be ca) Ce- 3)" ste iS . . HL )= ie me, = &. me , a , CO. AN By . ior. coeph irumnt 5 (x 3) = N>, ¥ (7 4). (y) n_-4 ] (n) n ’ £9? ~(D - GP CL £73) <n! foe nz _ ni gr ttl 3”? gr-3 Cn) ethimn se f(g)zo, n= eyes b) (6 pts.) If fis the same function as in part (a) above, find the Taylor series of the derivative (x) about a = 3. We get an [prrerh ode [pa ors Lipp = & Labatt) eos) fis2 grr 6 8x9 is oS 9 5. (26 pts.) Let dy, @n be an infinite series of positive terms, and S, its sequence of r 7 » partial sums. Suppose that we are given that lim)... = = 2 ze (a) Does the series pan a, converge or diverge? Justify your answer. 2a S Ge Gn =e y dim = Sr? “appa a n oye. Then Avr n ye p oe a ee a comlrnne Thr du 4 on 4, ) dan = = O: - = Oo h+aa Vb bois Pinu te ame divnyger (b) Is {a,} a bounded sequence? Justify your answer. Stree rine divoger, frm Sp = - 4 4 be bounded, Poe 7) % We would tayre oc fe ce UB Y Sim we. tire Y 20 ie wriddl gt dann $2 2 ¢ vonbang heypetbion hn vo rth bounded - Sa (c) Find limp an sin( ry )y a ABTA Oy te eb c a0-(65)% Str = “4 = = >t nore, ae om BE eddies ee “ O so ne Sin ‘) y a> *G Sn 5 a barn by sn Z— > dim a ae =o NY) eS. ak naT dba bet — (d) Give an example of an infinite series satisfying the condition in part (a). Tishen Ube Aone =~