exo6.mw

> Exercice 6

> restart:Digits:=30:N:=100:

Grace à l'inégalité de Taylor-Lagrange

> abs(f(1) -Sum(Diff(f(0),t$k)/(k!),k=0..'N'))<=sup(abs(Diff(f(t),t$(k+1))))/((n+1)!);

abs(f(1)-(Sum((Diff(f(0), `$`(t, k)))/factorial(k), k = 0 .. N))) <= sup(abs(Diff(f(t), `$`(t, k+1))))/factorial(n+1)

On etudie donc la suite de Taylor (Taylor_n) associée à la fonction t->4/(1+t^2)

> Taylor[n]:=4*Sum((-1)^k/(2*k+1),k=0..n)=Taylor[n-1]+4*(-1)^n/(2*n+1);

Taylor[n] := 4*(Sum((-1)^k/(2*k+1), k = 0 .. n)) = Taylor[n-1]+4*(-1)^n/(2*n+1)

> Taylor:=proc(N) options remember;
local S;
if N=0 then S:=4; else
S:=4*(-1)^N/(2*N+1)+Taylor(N-1); fi;
S;
end:

>

> for i from 0 to N do 4*Sum((-1)^k/(2*k+1),k=0..i)=evalf(Taylor(i)), erreur=evalf(abs(Taylor(i)-Pi)); od;

4*(Sum((-1)^k/(2*k+1), k = 0 .. 0)) = 4., erreur = .85840734641020676153735661672

4*(Sum((-1)^k/(2*k+1), k = 0 .. 1)) = 2.66666666666666666666666666667, erreur = .47492598692312657179597671661

4*(Sum((-1)^k/(2*k+1), k = 0 .. 2)) = 3.46666666666666666666666666667, erreur = .32507401307687342820402328339

4*(Sum((-1)^k/(2*k+1), k = 0 .. 3)) = 2.89523809523809523809523809524, erreur = .24635455835169800036740528804

4*(Sum((-1)^k/(2*k+1), k = 0 .. 4)) = 3.33968253968253968253968253968, erreur = .19808988609274644407703915640

4*(Sum((-1)^k/(2*k+1), k = 0 .. 5)) = 2.97604617604617604617604617605, erreur = .16554647754361719228659720723

4*(Sum((-1)^k/(2*k+1), k = 0 .. 6)) = 3.28373848373848373848373848374, erreur = .14214583014869050002109510046

4*(Sum((-1)^k/(2*k+1), k = 0 .. 7)) = 3.01707181707181707181707181707, erreur = .12452083651797616664557156621

4*(Sum((-1)^k/(2*k+1), k = 0 .. 8)) = 3.25236593471887589534648358178, erreur = .11077328112908265688384019850

4*(Sum((-1)^k/(2*k+1), k = 0 .. 9)) = 3.04183961892940221113595726599, erreur = 0.9975303466039102732668611729e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 10)) = 3.23231580940559268732643345646, erreur = 0.9072315581579944886379007318e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 11)) = 3.05840276592733181776121606516, erreur = 0.8318988766246142070142731812e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 12)) = 3.21840276592733181776121606516, erreur = 0.7681011233753857929857268188e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 13)) = 3.07025461777918366961306791701, erreur = 0.7133803581060956884957546627e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 14)) = 3.20818565226194229030272308943, erreur = 0.6659299867214905184007970615e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 15)) = 3.07915339419742616127046502491, erreur = 0.6243925939236707719217835837e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 16)) = 3.20036551540954737339167714612, erreur = 0.5877286181975413492903376284e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 17)) = 3.08607980112383308767739143184, erreur = 0.5551285246596015078525195144e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 18)) = 3.19418790923194119578549953994, erreur = 0.5259525564214795732285615666e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 19)) = 3.09162380666783863168293543738, erreur = 0.4996884692195460677970794590e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 20)) = 3.18918478227759472924391104714, erreur = 0.4759212868780149078126766386e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 21)) = 3.09616152646364124087181802388, erreur = 0.4543112712615199759082535940e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 22)) = 3.18505041535253012976070691277, erreur = 0.4345776176273689129806352949e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 23)) = 3.09994403237380672550538776383, erreur = 0.4164862121598651295725561945e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 24)) = 3.18157668543503121530130613118, erreur = 0.3998403184523797683866274790e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 25)) = 3.10314531288601160745816887628, erreur = 0.3844734070378163100447450700e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 26)) = 3.17861701099921915462798019703, erreur = 0.3702435740942591616533681375e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 27)) = 3.10588973827194642735525292431, erreur = 0.3570291531784681110739045897e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 28)) = 3.17606517686843765542542836290, erreur = 0.3447252327864441696278497962e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 29)) = 3.10826856669894613000169954934, erreur = 0.3332408689084710846094383394e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 30)) = 3.17384233719074940869022413951, erreur = 0.3224968360095617022758075623e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 31)) = 3.11035027369868591662673207601, erreur = 0.3124237989110732183591130727e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 32)) = 3.17188873523714745508827053755, erreur = 0.3029608164735421662562715427e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 33)) = 3.11218724269983402225244964203, erreur = 0.2940541088995921621019374125e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 34)) = 3.17015825719258764544085543913, erreur = 0.2856560360279440697821205585e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 35)) = 3.11382022902357356093381318561, erreur = 0.2777242456621967752883019767e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 36)) = 3.16861474957151876641326524041, erreur = 0.2702209598172552795062185713e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 37)) = 3.11528141623818543307993190707, erreur = 0.2631123735160780538271147621e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 38)) = 3.16722946818623738113187995902, erreur = 0.2563681459644414266923657574e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 39)) = 3.11659655679383231784074071851, erreur = 0.2499609679596092062190266477e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 40)) = 3.16597927284321503389012343456, erreur = 0.2438661925342179542748005128e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 41)) = 3.11778650175887768449253307312, erreur = 0.2380615183091555397011031016e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 42)) = 3.16484532528828944919841542606, erreur = 0.2325267169849621073577204278e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 43)) = 3.11886831379403657563519703525, erreur = 0.2272433979575666282744634803e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 44)) = 3.16381213401875567675879254087, erreur = 0.2221948042896243829614915759e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 45)) = 3.11985609006271172071483649692, erreur = 0.2173656352708151774780688636e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 46)) = 3.16286684275088376372558918509, erreur = 0.2127418916109052526294580181e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 47)) = 3.12076157959298902688348392193, erreur = 0.2083107399680421157915946135e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 48)) = 3.16199869299505088255358701471, erreur = 0.2040603940525764409094363143e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 49)) = 3.12159465259101047851318297431, erreur = 0.1999800099878275994946040897e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 50)) = 3.16119861298705008247357901391, erreur = 0.1960595939725684401093563063e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 51)) = 3.12236366153073940286192852848, erreur = 0.1922899205905383560071485480e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 52)) = 3.16045889962597749810002376657, erreur = 0.1886624603618425963738038329e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 53)) = 3.12307572205588404015609853293, erreur = 0.1851693153390919830654485035e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 54)) = 3.15977296976230605850472238614, erreur = 0.1818031617251282004207900286e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 55)) = 3.12373693372627002246868635010, erreur = 0.1785571986352321599395703318e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 56)) = 3.15913516381476559768992528816, erreur = 0.1754251022497235922728190488e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 57)) = 3.12435255511911342377688180989, erreur = 0.1724009847067981468576157339e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 58)) = 3.15854058930714761181106984408, erreur = 0.1694793571735437334842646080e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 59)) = 3.12492714392899635130686816341, erreur = 0.1666550966079688715577521987e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 60)) = 3.15798499516866577279447146920, erreur = 0.1639234157887253433182808592e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 61)) = 3.12546466996541374027414626594, erreur = 0.1612798362437949818849711734e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 62)) = 3.15746466996541374027414626594, erreur = 0.1587201637562050181150288266e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 63)) = 3.12596860697328775602217776201, erreur = 0.1562404661650548244046562127e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 64)) = 3.15697635891127225214620876976, erreur = 0.1538370532147901368356538648e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 65)) = 3.12644200776623408420727747205, erreur = 0.1515064582355915425536591123e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 66)) = 3.15651719573615889623735266002, erreur = 0.1492454214636565777470927674e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 67)) = 3.12688756610652926660772303039, erreur = 0.1470508748326397185492035289e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 68)) = 3.15608464639850006952743105959, erreur = 0.1449199280870683106478767631e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 69)) = 3.12730766798123388247707134736, erreur = 0.1428498560855935598557203592e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 70)) = 3.15567646230747501722884439700, erreur = 0.1408380871768177876620101372e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 71)) = 3.12770443433544704520087236903, erreur = 0.1388821925434619326177101425e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 72)) = 3.15529064123199876933880340351, erreur = 0.1369798764220553087616002023e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 73)) = 3.12807975687825727274016394773, erreur = 0.1351289671153596572247943555e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 74)) = 3.15492539446214989018982837726, erreur = 0.1333274087235665172718499398e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 75)) = 3.12843532823698432727592109249, erreur = 0.1315732535280891118672229079e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 76)) = 3.15457911908665752989030017746, erreur = 0.1298646549686429142765679418e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 77)) = 3.12877266747375430408384856456, erreur = 0.1281998611603893437879481872e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 78)) = 3.15425037448012373083544092124, erreur = 0.1265772089033049237279753796e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 79)) = 3.12909314177572121511217048099, erreur = 0.1249951181407202335047290229e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 80)) = 3.15393786227261562505005867975, erreur = 0.1234520868282238658741529647e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 81)) = 3.12939798497200212811754334233, erreur = 0.1219466861779111034510004095e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 82)) = 3.15364040921442637054178576657, erreur = 0.1204775562463313207914238329e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 83)) = 3.12968831340604313700885163483, erreur = 0.1190434018375010145379174845e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 84)) = 3.15335695245929757487867412004, erreur = 0.1176429886950433641603073676e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 85)) = 3.12996513959380049885528230717, erreur = 0.1162751399599273960736107611e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 86)) = 3.15308652687703749307493548636, erreur = 0.1149387328724425461229210308e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 87)) = 3.13022938401989463593207834351, erreur = 0.1136326956989860253056503977e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 88)) = 3.15282825407639181107332128136, erreur = 0.1123560048659857261067789808e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 89)) = 3.13048188536130801219063971711, erreur = 0.1111076822848522627200366617e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 90)) = 3.15258133287512016688677231380, erreur = 0.1098867928532692842412893052e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 91)) = 3.13072340937785240732393078374, erreur = 0.1086924421194083113871259954e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 92)) = 3.15234503099947402894555240537, erreur = 0.1075237740968079048290902209e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 93)) = 3.13095465666792322680651497221, erreur = 0.1063799692187001165612841107e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 94)) = 3.15211867783194439082767899338, erreur = 0.1052602424215115236503561010e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 95)) = 3.13117626945498104004233867924, erreur = 0.1041638413481219842030470404e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 96)) = 3.15190165805601730947239049271, erreur = 0.1030900446622407100974710943e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 97)) = 3.13138883754319679665187767220, erreur = 0.1020381604659644181076571108e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 98)) = 3.15169340607111557837776599707, erreur = 0.1010075248132233991512261379e-1

4*(Sum((-1)^k/(2*k+1), k = 0 .. 99)) = 3.13159290355855276430741423828, erreur = 0.999975003124047415522914500e-2

4*(Sum((-1)^k/(2*k+1), k = 0 .. 100)) = 3.15149340107099057525268787012, erreur = 0.990074748119733679004448684e-2

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