MATH 201 Final, Fall 2013

1. (8 points) Find the limit of the following sequence a, = (e” — 1)*. Uh, D0 o oC) 2 OC d., wer et Spl | Dy One 2. Determine if the folk ge. Justify your answers a. (point) Y> +, 3) ev p~terise wi ke er\ Sanvn n b.(@poins) 5 (1-2) —) é 42 2 dou nat a ley ate tera F real « (8 points) 5 (~ 1" sinn eel AST because Ann 1400 aluays fide. we rae § o, z& aie prerces p2ed) [a ones a8 : +(e) + ee \ \ _ (rely by ms ke 2 sy =O Ey” ee)” wy 3. (16 points) Find the sum of the series > point + = 1 COG @FD@+2) \ rel ecopef- pee Fi Live’ at peries ; =n 2 (2 | tog Vt: = —s 1+% te (lyn 4, (2 points) What is the interval of convergence of the power series > fre +2)", m0 wet 3n+l = es & (be sure to check convergence at the endpoints). | Bvt =z fp («49 On Bw zt) ae Pee kei L) (-t Ess aes eG 2 Me es “a gn +h AST 4ev ave AQ. 2 dink ment 3 nel gre +o) (4) 5. The Maclaurin series for tan z is given by: euzkee y= x. 3 15 a. (8 points) Using the series, find the first three nonzero terms in the Maclaurin series for §(2) = Incos'z). . (hint: whatis pa). FOO) 2 BAN Ko Ae og Cos % 4 x 6 or q aS 3 b. (6 points) For what values of can we replace sin x by x — = with an error of magnitude no greater than 107%, pinks xo, 2(% ae ‘ nel 9 LG) EPO ¢ is gt = S » . < (0s YL Salo xl C5200 4 6. (6 points) Use power series to evaluate the limit: lin a a @ = (4%1 8%, —-~— a ne a ak Cn ete. a na oe -1 =z 1) 2) 3) 4) MATHEMATICS 201 Quizt Investigate the convergence or divergence of : a) a, = (-1" sine” b), 4, =(0+ ba 9 Ses 4) a,=vn sin= (-1)" B Chim n i S(- non, Find S for 2 1)* 2 ers Far f= T3 apply taylor series for x = 1 Find the radius of convergence of the following series : (nl? ors Onl ) b) 4) cheg <i = lima, 0 because sine” <1 n n a,=(1+-)* 4an+5 ar ga jim@m =e 4 ts oct a, 3742" 4 4a +5 => converges ie 1 a,= nein afasin4, lim—7- =1 but at. converges by integral test n we? eal => Vnsin— converges n 1)", ; y ee 7 it converges by leibneiz theorem . it also converges absolutely by the integral test . 2) - © 4k yt 4, x y s Lee i 4y -4(_ 1 - 4 ,) 2 se 3 al a 1 1 1 Lies) mei 3) 3 2+3x 1 1-2/3 12 en ay" thee 3 aT) 4) Gay +x _ a, (2n+l)(Qn+2) 4 —4<x<4 at both endpoints there is divergence a (al? okt x 2n! 2 lim 2ule 4(n +1) ntl) _2nt20, diverge a, @n+h@n+2) (2n+l) n+l Ww ~ Ly) 2) 3) 4) MATHEMATICS 201 (Ist Semester, . + _-+) -QUIZ1 “ Do the following sequences converge or diverge ? a) ad- ha bh d+" cm mORGy a) (1—cos2) n 3 (nt) calculate 2 1 7 2 n(n +1) 6) Ye a0 a)Show that if Sa, converges => ) a’, converges. b)Ea, div, a,>0 lima, 0 Investigate for convergence and divergence . x 135.....(2n- 1) » mi 2466......20 135....(2n-1) m » n(2.46....2n) 20 ° — z (in@n + ny Solution: In’n 1) a oe 2) lim,,.4, =lim,,.-2)"* =0--)"a+ Lym el yent . idl mew en vn Th e ) converges . 1 La? n(in inn)? b) 1 rk avn 1 w(1 + 4) 0 lim, ye (+4) =limnJn x w(i+4) = 35 Find the domain of convergence of ae undetermined, use hopital's rule a) nx)". er ~ . Inga +1)-Inn +i na 3 0% b) 2" > in—— = a ssn. - “ mt 3a Q ped =3) = 3)" => tim + a =e" =0 mo «64 4 a, sag Ow “oer = inet) * in( 2) a 15x16 ry ar a) 2 a) b) a, = r(1~ cos”) (ai- coe?) 0 lim,,.. 4, = lim 2 sin— lim, ,.4, = lim iz n 2.2 = cos— lim 7 N.=-2 = converges w 2 2? " P= 1+ —+—4+..4— a a n" ts => e">— =nal> a nt @ => (al)">2 > lim(@)"” lim? =0 e e => diverges 5 lise fn F 1 a sy td er -2, n+l n+1 s=lim,,..5,=1 ze -3(3) 2G) Yn i -(3) 1-(3) s=lims, = a a e n= rs undetermined apply hopital rule lim——2 apply hopital rule 4) Apply limit comparison test >Y—* | diverges sevee(2n— 1)(2.4.6...+--2) (2.4.66.....2n)? = 2n! a1.4(n!)? aga _ n+ I)2n+2) ig a, (n+)? =lim on RS Si n(2.4.6....20) mim sit diverges by integral test 135.....(24-1) 4: > Yo — diverges x n(2.4.6......27) e ret 7 compare it with b, ="? jim 2 =0 nw Inn rm diverges 2 y diverges 5) b) 1 x nln? (Ininny'? nin®? n(ninn)"? is decreasing positively => lima,>0 1 2 gs integral test [————————5 = (InInn) diverge gral Sam" in(ininny” \ Ydinx)" = Ylnx(inz)™ m0 minx Inx) 1-Inx converges if -1<Inx<1 ml/ie<x<e end points for 1/e it diverges for e it also diverges > = = qn o<2<l 2% >0 Vx -oSxs0 end points forx=0 >i converges forx = -0© x() =0 converges ee -3) 3)" a a 4 acre) el ee both endpoints converge Bikic) MATHEMATICS 201 (Ist Semester, QUuIZ1 1) Study the convergence or the divergence of : o ay a) ue) n=O\ a3 2n+3 » $ 3 Ol (an42 2 (n+Jn) when a) = ~ On ee n+l” (asada ae 2) f@)=F 7 Gz Find Taylor's series for fat a= -1 and Ci 3) Find S for a) 2 ( ! Pa b) ~ iJ Letiedn " 4 1 x+l = » al a Find domain of convergence a) Absolutely, conditionally by) Atx=-4 n n n+2 1 n+ a) |—] =(-—) er n+ n+3. Hae 1 5) Find iF with an error of less than 0.1 nt: x nt3 1 (4) nt 1 ane . (an+2)" 7 lim9 = 5) tim Feet tim 5° tne = ~ + ne no 4, (anas) 3 3n+5\ 3n4 e . A 9 3 19)! 1 9 1. = lim 3) stim ~1.| = tim ——<~=0 converse =e 2) 3n+5| 1 mel sian +5 nfo 5 for n> 81 = Limb > 145417 % ant] > ag4#0 °) but ag) > 0=> Lima, , >) *° ‘Then U,, diverge 3 ra ;