Math 2414 Activity 16 (Due by end of class August 13) 1. Let f be a positive, continuous, decreasing function for x 1, and suppose that

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1 Mth Actvty 6 (Due y ed of clss August ). Let f e ostve, cotuous, decresg fucto for x, d suose tht f. If the seres coverges to s, d we cll the th rtl sum of the seres the the remder doule equlty r 0 s, stsfes the r f x dx. Sce f x r s s, we lso get tht 0 s s f x dx whch mles tht. Ths s equvlet to s s s f x dx f x dx. Use ths lst equlty to roxmte the sum of the followg coverget seres usg the dcted umer of terms, d gve estmte of the mxmum error. ) (sx terms) ) (four e. Use the result of rolem, followg coverget seres: terms) r f x dx, to fd the smllest so tht r.00 for the ) )

2 . Prolem gve us uer oud for the remder, fd lower oud. r, ow we ll From the fgure, you c see tht r f xdx. Use ths result to fd wht the mmum error s whe you use the tructed seres the two rts of rolem. Use the Itegrl Test to determe covergece or dvergece for the followg seres: e 7. 0 t 8. l 9. e 0.. l l l

3 . ) Suose tht f s ostve, cotuous decresg fucto for x, g s cotuously dfferetle fucto defed for x d lm g x g x f x s ostve d x evetully decresg, the show tht the two seres ether oth coverge or oth dverge. {Ht: From the tegrl test, we hve tht. If f d g f g f d f x dx ether oth coverge or oth dverge, d tht g f g d coverge or dverge. But g x f g x dx ether oth f g xgxdx f udu, so tke t wy.} g ) Use rt ) to determe covergece or dvergece of the seres c) Use rt ) to determe covergece or dvergece of the seres A fte roduct. l f x g x x {Ht: :Let d l {Ht: :Let x.} ll. d l f x g x l x x.} coverges f oly ftely my of ts fctors re zero d the sequece of rtl roducts(omttg the zero fctors),, coverges to o-zero umer. Assumg tht ll of ts fctors re ostve, ts covergece c lso e determed y cosderg the fte seres l l coverges to s, the s e.. If

4 . Show tht dverges. {Ht: Exme l, wth.}. Show tht l lm, or use the fct tht, sce coverges, d fd ts vlue. {Ht: ll l l l, so l l l l l, or, } 5. Show tht coverges, d fd ts vlue. 5 ; odd 5 ; eve 5 5 ; odd 5 5 ; eve 5

5 6. Show tht coverges, d fd ts vlue ) For the fte roduct x 0, use ducto to show tht the rtl roduct x, for x x x x x 0 x {Ht: For 0, the left sde s x, d the rght sde s x tht the formul s true for, x x x x. x x x x. Assumg } ) If x, show tht the rtl roduct x x x x x 0 c) Uder wht codtos o x wll the fte roduct x wll t coverge to? {Wtch out for x.} coverge, d wht d) Use the revous rt to fd the vlue of the fte roduct 0. 0

6 8. Show tht coverges, d fd ts vlue. 5 7 {Ht:,,, so t looks lke the umertors re odd umers strtg t, d the deomtors re odd umers strtg t. As formul, t looks lke. To rove t y ducto, t s true for, ssumg t s true for, the.} 9. Show tht 0 0 coverges, d fd ts vlue. {Ht: } 0

7 0. The rocedure egs wth crcle of rdus. A equlterl trgle s scred, the crcle s scred, the squre s scred, the other crcle, the regulr etgo, other crcle, the regulr hexgo, other crcle,. The questo s wht hes? Do the closed fgures cotue to dmsh utl they vsh to ot? Do they coverge to closed curve, d f so wht s t? Let s look t the sequece of the olygol rd: cos cos

8 cos, cos cos cos cos cos cos, cos cos, cos cos cos 5 cos cos cos 5 cos cos 5

9 So the sequece of olygol rd s gve y cos, cos cos, cos cos cos, cos cos cos cos,, where s the 5 5 umer of sdes of the regulr olygo. If lmcos cos cos 5 cos exsts, the the lmtg curve wll e crcle wth ths lmt s ts rdus. So we eed to vestgte the fte roduct cos or equvletly, cos. Ths fte roduct coverges f the ssocted l x seres l cos coverges. Sce lm, we oly eed to check the x0 x covergece of the seres cos. Comute cos lm to determe f the fte roduct cos coverges to the rdus of lmtg crcle. Ths rdus s clled the Keler-Bouwkm costt.. For the fte roduct, the relted seres s l. Use the fct tht l l l to fd the sum of the seres, d exoette to fd the vlue of the fte roduct. Or fd t drectly. Cuchy Codesto Test. Cosder the seres sequece of ostve umers. If s, the s where s o-cresg So f the seres coverges the so does.

10 O the other hd, s 5 8 Ths mes s For the seres, so f dverges the so does,, whch dverges... Use the Cuchy Codesto Test o the followg seres:(you c use t more th oce.) ) l ) l ll. Use the Cuchy Codesto Test to fd the vlues of c for whch the followg seres re coverget: (You c use t more th oce.) ) c l ) l l l. The ssumtos usg the Cuchy Codesto Test re mortt. Cosder the seres All of the terms re zero excet 8 6 the terms the -th osto, where the vlue s. Try the Cuchy Codesto Test o ths seres d commet o the result. 5. Ths rolem shows tht you hve to e creful lyg the tegrl test: ) Determe the covergece or dvergece of the mroer tegrl s x dx. {Ht: s x cosx ) Determe the covergece or dvergece of the fte seres s c) Do rts ) d ) cotrdct the Itegrl Test? Why? Why ot? c..}

11 , where I Kummer s Accelerto Method, oe wrtes the seres whose sum we wt to roxmte d where comuted exctly d whose terms re so close to the s tht the seres coverges much fster th (More secfclly, f lm cc ; 0 ; 0 s coverget,, the of the followg seres s s seres whose sum c e d so c e roxmted wth fewer terms. coverges to kow vlue, B, d cb c.) Aly ths method to fd the sum of ech wth error of t most.0 y usg the suggested seres. Comre the umer of terms requred wth Kummer to the umer of terms requred wth the tegrl estmte. 6., 7. 8.,, 9. Fd ll the ostve vlues of for whch the seres l coverges. {Ht: For 0 l. l l l, e l

12 0. Although the Hrmoc Seres dverges to fty, t does so very slowly. You c show tht l l. l l Scetsts eleve tht the uverse s llo yers old. Suose tht you eg ddg the terms of the Hrmoc Seres llo yers go d tht you dded oe term er secod. Assumg 65 dy yer, you would hve dded out llo 5,9,58,000,000,000 terms. Determe the roxmte totl of these 5,9,58,000,000,000 terms usg the verge of the uer d lower l ouds.

13 . The rolem s to cross desert y jee. There re o sources of fuel the desert, d you cot crry eough fuel jee order to mke the crossg oe go. You do ot hve tme to estlsh fuel dums, ut you do hve lrge suly of jees d drvers, oe of whch you wt to lose. How c you get cross the desert usg the mmum mout of fuel? Here s soluto to the rolem: We wll mesure the dstce jee c trvel terms of tkful of fuel; oe jee y tself c trvel dstce of oe tkful. If two jees set out together, they should trvel for of tkful, the Jee trsfers of ts tkful to Jee, d returs to se o the remg tkful. Jee s the le to trvel totl of tkfuls. Wth three jees, they should sto fter trvelg of tkful, the trsfer 5 5 of tkful from Jee to ech of Jees d, whch re ow full. Jee ow hs 5 of tkful, Jees d ow roceed s efore wth Jee returg wth emty tk to Jee. Betwee them, they hve eough fuel to get ck to the se. Mewhle, Jee hs trveled totl of 5 tkfuls. The sme resog shows tht wth four jees you c cheve dstce of 5 7 tkfuls, d wth jees you c get jee cross desert tht s tkfuls wde But we hve from the equlty the revous rolem tht l l d l l. Puttg them together, we get tht l l 5 7 l l Or more smly l 5 7 l. If jee c trvel 00 mles o oe tkful, d the desert s 900 mles wde, estmte the umer of jees requred to cross the desert usg the verge of the uer d lower ouds l For wht vlues of do the followg seres coverge? ) ) {Ht: commo deomtor.}

14 Determe covergece or dvergece for the followg seres: s s cos t {Ht: lmt comrso wth } {Ht: lmt comrso wth } {Ht: lmt comrso wth } cos x x {Ht: From Actvty5 #, cos x, you c show 0 cos x. } l l {Ht: For 0.. l l l {Ht: For. l. l l {Ht: l for e.} e l e, l e l e e e, ll.} l e l l {Ht: l.} l l.}

15 . We kow tht the sum of the recrocls of the coutg umers, dverges to fty. Wht f we oly dd the recrocls of those coutg umers tht do t hve the dgt 6? There re 8 oe-dgt coutg umers wthout 6:,,,,5,7,8, There re 7 two-dgt coutg umers wthout 6: 0,,,,,5,7,, There re 68 three-dgt coutg umers wthout 6: 00,0,0,0,0,05,07,, So o 6 dgt ) Use the revous dscusso to fd whole umer tht ouds the sum of the recrocls of whole umers wthout dgt of 6. Here re the mssg dgt sums: Mssg Dgt Sum (to 5 lces) ) Fd whole umer oud o the sum of the recrocls of the coutg umers wthout y eve dgts. There re 5 oe-dgt coutg umers wthout y eve dgts:,,5,7,

16 There re 5 two-dgt coutg umers wthout y eve dgts:,,5,7,9,,,, There re 5 three-dgt coutg umers wthout y eve dgts:,,5,7,9,,,, So o eve dgt c) Determe covergece or dvergece of the sum of the recrocls of the coutg umers tht hve oes-dgt of 6? There s oe-dgt coutg umer wth oe s dgt of 6: There re 9 two-dgt coutg umers wth oe s dgt of 6: 6,6,6,6,56,66,76,86, There re 90 three-dgt coutg umers wth oe s dgt of 6: 06,6,6,6,6,56,66,, So oe's dgt of 6 d) Determe covergece or dvergece of the sum of the recrocls of the coutg umers tht do t hve oes-dgt of 6?

17 Exteded Lmt Comrso Test: Suose d If lm 0, the 5. Suose tht s If lm, the re coverges f dverges seres of ostve terms. f coverges. dverges. coverges d 0. Determe the covergece or dvergece of usg the Exteded Lmt Comrso Test wth. 6. Use the Exteded Lmt Comrso Test to determe the covergece or dvergece of the followg seres: l l ) ) e e c) d) l 7. ) If 0 d lm sy out? {Ht: ) Wht f L 0? lm l L, where L s ether ostve umer or, the wht c you lm d Lmt Comrso or Exteded Lmt Comrso.} {Ht:.} l d

18 8. Suose tht coverges d ) If for ll N, the show tht,, d c re seres wth ostve terms, d tht c dverges. coverges. N N N {Ht:, so N N. N N N N N N N N N N N so multlyg gves N N N N N whch mes tht N N, so kee gog.} N N N N N c ) If for ll N, the show tht c 9. (Loglog Test) Suose tht 0 for d ) Wht hes f L? {Ht: Ths mes tht evetully l l l d y exoettg, you get tht kow out the seres ) Wht hes f L??} l {Ht: Ths mes tht evetully l ll l dverges. l lm l l L (ossly ). whch mles tht l l l l l l l or tht,. Wht do you l, where whch mles tht, d y exoettg, you get tht. Wht do you kow out the seres or tht l for l?}

19 50. Cosder the sequece of rtl sums of the hrmoc seres, s. Determe f the seres s s coverget or dverget. {Ht: s, whch mles tht.} s 5. Cosder the sequece of rtl sums of the geometrc seres, s Determe f the seres {Ht: s s coverget or dverget. s } 5. The Ressce mthemtc Petro Megol roved the equlty for x. ) Prove the equlty usg lger. {Ht: The equlty s equvlet to x x x x x x, d. } x x x x x x x ) Use the equlty rt ) to show tht the hrmoc seres dverges. {Ht: Suose tht to get tht tht coverges. Now grou ts terms fter the frst oe to threes. But the equlty rt ) mles.} 5. Let s exme the seres l, 0. ) Sce l xx for x 0 from Actvty, rolem #, susttute t to rrve t l t. ) Use the revous equlty to get x t wth t, 0 l. So wht vlues of wll gurtee covergece of the seres? t {Ht: Strt y relcg t wth l t.}

20 5. Cosder the sum x, where d re sequeces of rel umers. It s qudrtc olyoml x tht s oegtve. If we exd the sum, we get x x x x. From the qudrtc formul, we get tht f Ax Bx C 0, the t s dscrmt, B AC, must e less th or equl to zero. ) Set-u ths dscrmt equlty wth our rtculr qudrtc d fd equlty volvg ) Suose tht, show tht the seres,. s coverget seres wth 0. Use the revous equlty to coverges f. c) Show tht f, the ew seres my e dverget y cosderg d) If. l s coverget seres of o-egtve terms, the wht out 55. (Log Test) Suose tht 0 for d ) Wht hes f L? {Ht: Ths mes tht evetully exoettg, you get tht ) Wht hes f L? {Ht: Ths mes tht evetully l lm l l L(ossly ). l whch mles tht l l?. Wht do you kow out the seres l l l, d y exoettg, you get tht out the seres for?} l, d y?}, where whch mles tht. Wht do you kow

21 Use the Log Test of rolem #55 to determe covergece or dvergece of the followg seres: 56. l 57. l 58. l l Use the Loglog test of rolem #9 to determe covergece or dvergece of the followg seres: 59.!! Let s vestgte the fte roduct the rtl roduct k k 6.!! k. Notce tht 9 6 P. Ths c e wrtte s l k k k wth 9 6 l l l l l 9. e e e So the fte roduct s covergece or dvergece deeds o the covergece or k. l k dvergece of the seres k ) Show tht the fte roduct coverges. k l k ) Use estmte for the vlue of k roduct. 6. We kow tht the seres fty. From the Me Vlue Theorem wth f x to wth.0 to estmte the vlue of the fte dverges to fty. Let s fd the rte tht t dverges to x d k, we get tht k k k k for some c k c k. Ths smlfes to k k, d sce, we get tht k k c c k k. Smlrly, from the Me Vlue Theorem wth f x k k k k c x d k, we get tht for some k c k. Ths smlfes to

22 k k, d sce c c, we get tht k k k k. Puttg oth of these results together, leds to k k k k. k ) Use the revous result to fd uer d lower ouds o the th rtl sum of the seres, s. k k ) Use the revous result to fd whole umer uer d lower ouds o the,000,000 th rtl sum 6 0 k k. 6. The rme umers re those ostve tegers greter th tht re dvsle y oly d themselves(,,5,7,, ). A very fmous theorem sttes tht the sequece of rme umers,,5,7,, hs the roerty tht lm. Use the Lmt l Comrso Test to determe f the seres of recrocls of the rme umers, 5 7, s coverget or dverget. 65. The Srl of Theodorus(crc 50 B.C.) s formed from rght trgles s the cture. Strtg wth sosceles rght trgle wth legs of ut, ddtol rght trgles re formed wth oe leg lso of ut d the other leg equl to the recedg hyoteuse. The sequece of hyoteuses s,,, 5,. Wll the srl come to sto, or wll t cotue forever? I other words, does the sum of the teror gles coverge or dverge? {Ht: t, so t.}

23 66. The Srl of Hortous(crc 0 A. D.) s formed from rght trgles s the cture. Strtg wth sosceles rght trgle wth legs of ut, ddtol rght trgles re formed wth oe leg of uts d the other leg equl to the recedg hyoteuse. The 5 sequece of hyoteuses s,,,, 8. Wll the srl come to sto, or wll t cotue forever? I other words, does the sum of the teror gles coverge or dverge? {Ht: t ;, so t t t t.} 67. If the seres covergece or dvergece of {Ht: If d therefore out dverges, wth 0, the c you coclude ythg out the? dverges ecuse lm 0, the wht c you sy out lm, whch would mly tht here.},? O the other hd, f lm 0, the evetully,, so tke t from

24 68. Suose tht {Ht: s seres of ostve terms. Aother seres c e formed,, where,.e. the verge of the frst terms of the orgl seres. Does the seres 69. Suose tht the seres coverge or dverge?, so.} s coverget seres wth ostve terms. Determe f the followg seres must coverge, must dverge, or tht there s ot eough formto to tell. If t must coverge, show why. If t must dverge, show why. If there s ot eough formto to tell, gve exmle tht coverges d exmle tht dverges. ) d) ) c) e) f)