{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "War ning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 "\000" } {TEXT 258 27 "TP 1 : Des \351tudes de suites" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "1 Quatre vitesses de convergence" }}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 7 "Suite u" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "u:=n->if n=0 then 1. else a:=u(n-1):a-a^2/2 fi:" }}{PARA 7 "" 1 " " {TEXT -1 59 "Warning, `a` is implicitly declared local to procedure \+ `u`\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Deux remarques : \"1.\" au lieu de \"1\" pour avoir des flottants plutot que des fractions; s tockage de u(n-1) dans a pour ne pas le calculer deux fois " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(u(k),k=0..10);u(100);u(1 000);u(10000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\"\"\"\"!$\"+++++ ]!#5$\"++++]PF($\"++](o/$F($\"+PEq#e#F($\"+\"*\\=\\AF($\"+NLC'*>F($\"+ iR*pz\"F($\"+(fMbj\"F($\"+Efy,:F($\"+yy,*Q\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vf:z=!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+E!#7" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, (in u) too many levels o f recursion\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Il faudrait le c alculer avec une boucle !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "u:=proc(n)\nlocal r,k;\nr:=1.:\nfor k from 1 to n do r:=r-r^2/2 od :\nRETURN(r)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(u (k),k=0..10);u(100);u(1000);u(10000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\"\"\"\"!$\"+++++]!#5$\"++++]PF($\"++](o/$F($\"+PEq#e#F($\"+\"* \\=\\AF($\"+NLC'*>F($\"+iR*pz\"F($\"+(fMbj\"F($\"+Efy,:F($\"+yy,*Q\"F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vf:z=!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+E!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Il !y*>!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(1/u(10^k),k= 1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+$QJ$*>(!\"*$\"+Q!Q:K&!\" )$\"+*[>M/&!\"($\"+'p*[0]!\"'$\"+6Sm+]!\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "seq(ln(u(10^k)),k=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+e\"))R(>!\"*$!+fuMuRF%$!+3XDBiF%$!+G0H=&)F%$!+36*>3 \"!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq([%][k]-[%][k+ 1],k=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"+,$f.+#!\"*$\"+\\q!* [AF%$\"+?g.&H#F%$\"+_0i,BF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ln(10.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$4&e-B!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Connaissez-vous une fonction simpe telle que ln(f(10x))=ln(f(x))-ln(10) ? Moi oui..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Bonus : on demande \340 Maple un d\351veloppement asymptotique (le premier terme est un \351quivalent)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "asympt(rsolve(\{s(0)=1,s(n+1)=s(n)- s(n)^2/2\},s(n)),n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*&\"\"\"F%% \"nG!\"\"\"\"#*&,&%#_CGF%*&F(F%-%#lnG6#F&F%F'F%F&!\"#F%*&,,F%F%F+F%*&# F%F(F%)F+F(F%F%*&,&F+F(F(F%F%F-F%F'*&F(F%)F-F(F%F%F%F&!\"$F%*&,0#\"\"& \"\"$F%*&#F>F(F%F+F%F%*&#F>\"\"%F%F5F%F%*&#F%FDF%)F+F?F%F%*&,(F>F%*&F> F%F+F%F%*&#F?F(F%F5F%F%F%F-F%F'*&,&F>F%*&F?F%F+F%F%F%F9F%F%*&F(F%)F-F? F%F'F%F&!\"%F%-%\"OG6#*&F%F%*$)F&F>F%F'F%" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 7 "Suite v" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "v:= proc(n)\nlocal r,k;\nr:=1.:\nfor k from 1 to n do r:=arctan(r) od:\nRE TURN(r)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(v(k),k =0..10);v(100);v(1000);v(10000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$ \"\"\"\"\"!$\"+M;)R&y!#5$\"++vtdmF($\"+d<%Q(eF($\"+-^\"4J&F($\"+yL5#)[ F($\"+MZrTXF($\"+i]7!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^l%>(Q!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7K tC7!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(1/v(10^k),k=1 ..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+:<[LF!\"*$\"+%[(z+#)F%$\" +z-o#e#!\")$\"+RN/l\")F*$\"+))o)>e#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(ln(v(10^k)),k=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+]hd05!\"*$!+/9B/@F%$!+@GT^KF%$!+_rW-WF%$!++%HPb&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq([%][k]-[%][k+1],k=1.. 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"+a_l)4\"!\"*$\"+<9=Z6F%$\"+J V.^6F%$\"+[AG^6F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ln(10. )/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+YDH^6!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "asympt(rsolve(\{s(0)=1,s(n+1)=arcta n(s(n))\},s(n)),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(-%%sqrtG6# \"\"$\"\"\"-F&6#\"\"#F)-F&6#*&F)F)%\"nG!\"\"F)#F)F,*&,&%#_CGF)**#F(\"# !)F)F%F)F*F)-%#lnG6#F0F)F)F))F/#F(F,F)F)-%\"OG6#*&F)F)*$)F0F,F)F1F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "2/sqrt(6.);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+3e'\\;)!#5" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Suite w" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "w:=pr oc(n)\nlocal r,k;\nr:=1.:\nfor k from 1 to n do r:=sin(r)/2 od:\nRETUR N(r)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(w(k),k=0. .10);w(100);w(1000);w(10000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\" \"\"\"!$\"+C\\N2U!#5$\"+k&f@/#F($\"+at*R,\"F($\"+XIIh]!#6$\"+37dHDF/$ \"+C2lk7F/$\"+q]3Bj!#7$\"+o9_hJF6$\"++\"e2e\"F6$\"+5sy.z!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vvi%Q'!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bwM`v!$6$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?BvcS!%?I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(1/w(10^k),k=1..5);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+[i@l7!\"'$\"+q?Em:\"#@$\"+*G;RK\" \"$#H$\"+9g-lC\"%,I$\"+h*HVB\"\"&%4I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(ln(w(10^k)),k=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+M$)*H9(!\"*$!+sWi_p!\")$!+sqeLp!\"($!+KLoJp!\"'$!+eH \\Jp!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq([%][k+1]/[% ][k],k=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"+h=[L(*!\"*$\"+(R= E(**F%$\"+AVD(***F%$\"+d`s****F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Cherchons une fonction telle que ln(f(10x))=10ln(f(x))..." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "asympt(rsolve(\{s(0)=1,s(n+1 )=sin(s(n))/2\},s(n)),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#_CG \"\"\")#F&\"\"#%\"nGF&F&*(#F)\"\"*F&)F%\"\"$F&)F'F/F&F&*(#\"#u\"$v'F&) F%\"\"&F&)F'F6F&F&-%\"OG6#*&)F%\"\"'F&)F'F=F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Hop !" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "suite \+ z" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "z:=proc(n)\nlocal r,k; \nr:=1.:\nfor k from 1 to n do r:=sin(r)^2 od:\nRETURN(r)\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(z(k),k=0..10);z(100);z(1 000);z(10000);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"\"\"\"\"!$\"+$=M2 3(!#5$\"+!4J)HUF($\"+_'e\\o\"F($\"+O\">B\"G!#6$\"+&*R02z!#8$\"+&)*[@D' !#;$\"+$pO*3R!#A$\"+2'yz_\"!#M$\"+B'=ZL#!#f$\"+\\5\"4X&!$4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq(1/z(k),k=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+FHG79!\"*$\"+)fgTO#F%$\"+OP'[$fF%$\"+=UybN!\")$\"+E Npk7!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seq(ln(z(k)),k= 1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+D\\2_M!#5$!+A.B/')F%$!+oS %3y\"!\"*$!+G2;rNF*$!++^eUrF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq([%][k+1]/[%][k],k=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6& $\"+d2[#\\#!\"*$\"+?3tp?F%$\"+f\">`+#F%$\"+EQ2+?F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Cherchons une fonction telle que ln(f(x+1))=2ln(f (x))..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "asympt(rsolve(\{ s(0)=1,s(n+1)=sin(s(n))^2\},s(n)),n);" }}{PARA 8 "" 1 "" {TEXT -1 44 " Error, (in asympt) unable to compute series\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Dommage..." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 " 2 Un ph\351nom\350ne d'instabilit\351 num\351rique" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "1. Suite de Fibonacci" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 12 "F ormellement" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rsolve(\{f(n+ 2)=f(n)+f(n+1),f(0)=0,f(1)=1\},f(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&!\"\"\"\"\"*&#F'\"\"&F'-%%sqrtG6#F*F'F'F'),$*&F'F',&F'F'*$F+F 'F&F&!\"#%\"nGF'F1F&F'*(,&F2#F&F*F'F&F'),$*&F'F',&F'F'F2F'F&F3F4F'F;F& F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n=100,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&!\"\"\"\"\"*&#F'\"\"&F'-%%sqrtG6 #F*F'F'F',&F'F'*$F+F'F&!$,\"\"@w`?.n\\,%H#G-g]wE\"*(F1F',&F/#F&F*F'F&F ',&F'F'F/F'F0F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify (%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*&),&!\"\"F%*$-%%s qrtG6#\"\"&F%F%\"$,\"F%),&F%F%F*F%F/F%F)\"]p+Gdqh!p%e6SG3%*Q[B=\">Gq8( 3QdIW$Q&GKod$[W!R&poF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "n umer(%)/expand(denom(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"6v]\">E z\"[[Aa$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Je n'ai pas r\351ussi \340 faire autrement ! Je voulais \351viter le evalf..." }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 13 "Num\351riquement" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f:=proc(n)\noption remember;\nif n<=1 then n els e f(n-1)+f(n-2) fi;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"6v]\">Ez\"[[Aa$" }}}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "2. Suite g" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 12 "Formellement" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rsolve(\{g(n+2)=g(n)+3/2*g(n+1),g(0)=1,g(1)=-2\},g(n));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&)#!\"\"\"\"#%\"nG#\"\")\"\"&*&#\"\"$ F+\"\"\")F'F(F/F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n =100,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!gn.wki%4P`C_;*=&>(eDp:Zc mqUU>LN?07\"?s1!z3(=v'G&G]Kc%e\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+,O!fg( \"#?" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 13 "Num\351riquement" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "g:=proc(n)\noption remember; \nif n=0 then 1. elif n=1 then -2 else g(n-2)+1.5*g(n-1) fi;\nend:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+/O!fg(\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "T out va bien" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "3. Suite h" }} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 12 "Formellement" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rsolve(\{h(n+2)=h(n)+3/2*h(n+1),h(0)=-3,h(1)= 3/2\},h(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$)#!\"\"\"\"#%\"nG!\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(n=100,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##!\"$\"@w`?.n\\,%H#G-g]wE\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!+;FemB!#R" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 13 "Num\351riquement" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "h:=proc (n)\noption remember;\nif n=0 then -3. elif n=1 then 3/2 else h(n-2)+1 .5*h(n-1) fi;\nend;:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"n G6\"6#%)rememberGF(@'/9$\"\"!$!\"$F./F-\"\"\"#\"\"$\"\"#,&-F$6#,&F-F2F 5!\"\"F2*&$\"#:F:F2-F$6#,&F-F2F2F:F2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2 \")o*='\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "H\351h\351 !!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rsolve(\{h(n+2)=h(n)+3/2*h(n +1),h(0)=-3,h(1)=3/2\},h(n));" }}{PARA 8 "" 1 "" {TEXT -1 43 "Error, ( in h) too many levels of recursion\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Normal, h a maintenant une valeur... il faut changer de lettre , ou bien \"r\351initialiser\" h. C'est ce choix que je fais." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "unassign(h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rsolve(\{h(n+2)=h(n)+3/2*h(n+1),h(0 )=a,h(1)=b\},h(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%\"aG#\" \")\"\"&*&#\"\"%F)\"\"\"%\"bGF-!\"\"F-)#F/\"\"#%\"nGF-#F-F2*&,&F&#F/F) *&#F2F)F-F.F-F/F-)F2F3F-F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 518 "Si la condition h(0)=-2h(1) est v\351rifi\351e, on a une suite g\351om \351trique de raison de valeur absolue <1, donc qui tend vers 0. Mais \+ si cette condition n'est pas v\351rifi\351e, meme avec h(0)+2h(1) tr \350s faible, la puissance de 2 va faire exploser la solution. Si on f ait un calcul num\351rique, les erreurs d'arrondis font que, meme si a u d\351part h(0)+2h(1)=0, on va avoir h(N)+2h(N+1) tr\350s l\351g\350r ement diff\351rent de 0 \340 un moment donn\351, et \340 partir de l \340, la composante g\351om\351trique de raison 2 est non nulle, et fa it tout exploser..." }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "4. Une \+ suite de Fibonacci instable" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rsolve(\{i(n+2)=i(n)+i(n+1),i(0)=a,i(1)=b\},i(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**,&*$-%%sqrtG6#\"\"&\"\"\"\"\"$F*!\"\"F+,(%\"bG F+*&\"\"#F+%\"aGF+F+*&F'F+F/F+F+F+),$*&F+F+,&F+F+F&F-F-!\"#%\"nGF+F7F- #F-\"#5*,#F+F;F+,&F&F,F*F+F+,(F/F+*&F1F+F2F+F+F3F-F+),$*&F+F+,&F+F+F&F +F-F8F9F+FDF-F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "C'est le prem ier terme qui a une raison >1. On va donc faire en sorte d'annuler au \+ d\351part cette composante." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "rsolve(\{i(n+2)=i(n)+i(n+1),i(0)=-(1+sqrt(5))/2,i(1)=1\},i(n));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*$-%%sqrtG6#\"\"&\"\"\"!\"\"\"\"$ F+F*),$*&F*F*,&F*F*F%F*F+!\"#%\"nGF*F0F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(subs(n=100,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+g%*yU?!#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "i:=p roc(n)\noption remember;\nif n=0 then -(1+sqrt(5.))/2 elif n=1 then 1. else i(n-1)+i(n-2) fi;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+7pPva\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "BOUM" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "3 Des r\351currences du premier ordre non monotone" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "1. u[n+1]=exp(-u[n])" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->exp(-x):plot(\{f(x),x\},x=0.. 1);" }}{PARA 13 "" 1 "" {GLPLOT2D 285 113 113 {PLOTDATA 2 "6&-%'CURVES G6$7S7$$\"\"!F)F(7$$\"3dmmm;arz@!#>F+7$$\"3[LL$e9ui2%F-F/7$$\"3nmmm\"z _\"4iF-F27$$\"3[mmmT&phN)F-F57$$\"3BLLe*=)H\\5!#=F87$$\"3gmm\"z/3uC\"F :F<7$$\"3%)***\\7LRDX\"F:F?7$$\"3^mm\"zR'ok;F:FB7$$\"3w***\\i5`h(=F:FE 7$$\"3YLLL3En$4#F:FH7$$\"3qmm;/RE&G#F:FK7$$\"3\")*****\\K]4]#F:FN7$$\" 3$******\\PAvr#F:FQ7$$\"3(******\\nHi#HF:FT7$$\"3jmm\"z*ev:JF:FW7$$\"3 ?LLL347TLF:FZ7$$\"3,LLLLY.KNF:Fgn7$$\"3v***\\7o7Tv$F:Fjn7$$\"3'GLLLQ*o ]RF:F]o7$$\"3A++D\"=lj;%F:F`o7$$\"31++vV&RY2aF:Fbp7$$\"39mm;zXu9cF:Fep7$$\"3l******\\y))GeF:Fhp7$$ \"3'*)***\\i_QQgF:F[q7$$\"3@***\\7y%3TiF:F^q7$$\"36****\\P![hY'F:Faq7$ $\"3jKLL$Qx$omF:Fdq7$$\"3!)*****\\P+V)oF:Fgq7$$\"3?mm\"zpe*zqF:Fjq7$$ \"3%)*****\\#\\'QH(F:F]r7$$\"3GKLe9S8&\\(F:F`r7$$\"3R***\\i?=bq(F:Fcr7 $$\"3\"HLL$3s?6zF:Ffr7$$\"3a***\\7`Wl7)F:Fir7$$\"3#pmmm'*RRL)F:F\\s7$$ \"3Qmm;a<.Y&)F:F_s7$$\"3*F:7$F8$\"3K-P;0r(Q+*F:7$F<$\"3D!)p*>rcs#))F:7$F?$\"3\"3%\\ $=jE![')F:7$FB$\"3)zt'>wO\\m%)F:7$FE$\"3ku.Ey_L*G)F:7$FH$\"3zN,nmH(46) F:7$FK$\"3!zRI$zH0dzF:7$FN$\"3#z>E?vnsy(F:7$FQ$\"30E#=RVI/i(F:7$FT$\"3 z*z9KoMIY(F:7$FW$\"3]4E%faAHK(F:7$FZ$\"3'F:7$F\\p$\"3 g^8D36IogF:7$F_p$\"3<))yCcZ(f$fF:7$Fbp$\"3a&R.csNJ#eF:7$Fep$\"3PZ@H2Yn .dF:7$Fhp$\"3mPc7XF$Ge&F:7$F[q$\"3!4^uAY!4naF:7$F^q$\"3%)**[q&Q)Qd`F:7 $Faq$\"3\"og2v)*f\"Q_F:7$Fdq$\"3VQiAaHHL^F:7$Fgq$\"3#[%*o9VTO-&F:7$Fjq $\"3p%[DY/0j#\\F:7$F]r$\"3/Cq\"eNZ?#[F:7$F`r$\"38&>)[:Y'fs%F:7$Fcr$\"3 7NlK'Qwvi%F:7$Ffr$\"3\\^rwoWOLXF:7$Fir$\"399-:sxyOWF:7$F\\s$\"31FuI^%= dM%F:7$F_s$\"3PDU'R()>XD%F:7$Fbs$\"3@3T-]n'f;%F:7$Fes$\"3I=if9:A'3%F:7 $Fhs$\"3iz>6h+q'*RF:7$F[t$\"3_>3][QH=RF:7$F^t$\"33h,p<+ROQF:7$Fat$\"3 \")Hj)ye,'fPF:7$Fdt$\"3LBWr6WzyOF:-Fgt6&FitF(FjtF(-%+AXESLABELSG6$Q\"x 6\"Q!Fg^l-%%VIEWG6$;F(Fdt%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(f(x)=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+/HVrc!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "u:=n->if n =0 then 1. else f(u(n-1)) fi:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(u(k),k=0..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "67$\"\"\"\" \"!$\"+7WzyO!#5$\"+vi+ApF($\"+1]t/]F($\"+^`VigF($\"+gy&RX&F($\"+bL7'z& F($\"+9Y:,cF($\"+^6V6dF($\"+uMz[cF($\"+]sG%o&F($\"+Kt9kcF($\"+tjcvcF($ \"+>\"*3pcF($\"+ABwscF($\"+%)*y1n&F($\"+,0'=n&F($\"+,/>rcF($\"+S/drcF( $\"+-\\NrcF($\"+VrZrcF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 538 "Les s uites extraites d'indices pairs et impairs semblent adjacentes.\nD\351 j\340, elles v\351rifient des relations de la forme v[n+1]=f(f(v[n])), et fof est strictement croissante, donc la relation v[1]>v[0] se prop age : v est donc strictement croissante, et de meme w est strictement \+ d\351croissante. Elles sont par ailleurs localis\351es dans [0,1], d'o \371 les convergences. Leurs limites l1 et l2 sont solutions de l'\351 quation f(f(x))=x. Bien entendu, l'unique X tel que f(X)=X v\351rifie \+ \351galement f(f(X))=X, mais il pourrait y en avoir d'autres, a priori ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(\{f(f(x)),x\},x=0 ..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 345 169 169 {PLOTDATA 2 "6&-%'CURV ESG6$7S7$$\"\"!F)F(7$$\"3dmmm;arz@!#>F+7$$\"3[LL$e9ui2%F-F/7$$\"3nmmm \"z_\"4iF-F27$$\"3[mmmT&phN)F-F57$$\"3BLLe*=)H\\5!#=F87$$\"3gmm\"z/3uC \"F:F<7$$\"3%)***\\7LRDX\"F:F?7$$\"3^mm\"zR'ok;F:FB7$$\"3w***\\i5`h(=F :FE7$$\"3YLLL3En$4#F:FH7$$\"3qmm;/RE&G#F:FK7$$\"3\")*****\\K]4]#F:FN7$ $\"3$******\\PAvr#F:FQ7$$\"3(******\\nHi#HF:FT7$$\"3jmm\"z*ev:JF:FW7$$ \"3?LLL347TLF:FZ7$$\"3,LLLLY.KNF:Fgn7$$\"3v***\\7o7Tv$F:Fjn7$$\"3'GLLL Q*o]RF:F]o7$$\"3A++D\"=lj;%F:F`o7$$\"31++vV&RY2aF:Fbp7$$\"39mm;zXu9cF:Fep7$$\"3l******\\y))GeF:Fhp7 $$\"3'*)***\\i_QQgF:F[q7$$\"3@***\\7y%3TiF:F^q7$$\"36****\\P![hY'F:Faq 7$$\"3jKLL$Qx$omF:Fdq7$$\"3!)*****\\P+V)oF:Fgq7$$\"3?mm\"zpe*zqF:Fjq7$ $\"3%)*****\\#\\'QH(F:F]r7$$\"3GKLe9S8&\\(F:F`r7$$\"3R***\\i?=bq(F:Fcr 7$$\"3\"HLL$3s?6zF:Ffr7$$\"3a***\\7`Wl7)F:Fir7$$\"3#pmmm'*RRL)F:F\\s7$ $\"3Qmm;a<.Y&)F:F_s7$$\"322 RF:7$F5$\"3'=6x1n]e)RF:7$F8$\"3ZXfh*e?T1%F:7$F<$\"3'fyg@*)Ql8%F:7$F?$ \"3Qv4*z\\Y8@%F:7$FB$\"3j$R!zTW\\)G%F:7$FE$\"3T$*=8)oX^O%F:7$FH$\"3G]a pE>qVWF:7$FK$\"3)[`I!z%GE^%F:7$FN$\"3K#R^#p.!**e%F:7$FQ$\"3nSEo&R>rm%F :7$FT$\"3m-[P>!f6u%F:7$FW$\"3OMh0rf03[F:7$FZ$\"3!*)z8W.hr)[F:7$Fgn$\"3 (zW4`czP&\\F:7$Fjn$\"3g;a$o(pzI]F:7$F]o$\"3#H.qC3E&)4&F:7$F`o$\"3IL.# \\%oLs^F:7$Fco$\"3+7-\\H,7U_F:7$Ffo$\"3:$)Q0%Q#R9`F:7$Fio$\"30H7\"*=:E !Q&F:7$F\\p$\"3[%[w%Q'f2X&F:7$F_p$\"3\"[!)pa)oOBbF:7$Fbp$\"3[7/u$)\\/' e&F:7$Fep$\"3`\"ya[owJl&F:7$Fhp$\"3w)>w?:0>s&F:7$F[q$\"3BFbdKl^)y&F:7$ F^q$\"3L1h(*[tO_eF:7$Faq$\"3[c]f9=cAfF:7$Fdq$\"3nm8\\fn*\\)fF:7$Fgq$\" 3ztZx!G%)40'F:7$Fjq$\"3h%*)RT&*p,6'F:7$F]r$\"3TCj\\3k?uhF:7$F`r$\"3NJc k)y:QB'F:7$Fcr$\"3iei`9>X&H'F:7$Ffr$\"33QQAUG/bjF:7$Fir$\"31#=VB2:nT'F :7$F\\s$\"3HneKf&=aZ'F:7$F_s$\"3(H8p>cVZ`'F:7$Fbs$\"3C\"*3kJx'Gf'F:7$F es$\"3Zt88**GlXmF:7$Fhs$\"3]A'f0%GT0nF:7$F[t$\"3oLga\"4%>enF:7$F^t$\"3 ypf2)etP\"oF:7$Fat$\"3sO\"fl)oHmoF:7$Fdt$\"3$RY`bF1?#pF:-Fgt6&FitF(Fjt F(-%+AXESLABELSG6$Q\"x6\"Q!Fi^l-%%VIEWG6$;F(Fdt%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Il semblerait qu'il n'y ait qu'un \+ point fixe. Pour cela, on consid\350re la diff\351rence f(f(x))-x" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(f(f(x))-x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#,$%\"xG!\"\"\"\"\"-F&6#,$F%F*F+ F+F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "g:x->f(f(x))-x est str ictement d\351croissante, et g(0)>0>g(1), donc il existe un unique poi nt l tel que g(l)=0, c'est-\340-dire f(f(l))=l; gagn\351. " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "2. u[n+1]=exp(-3u[n])" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g:=x->exp(-3*x):plot(\{g(x),x\},x=0..1); " }}{PARA 13 "" 1 "" {GLPLOT2D 285 113 113 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$\"\"!F)F(7$$\"3dmmm;arz@!#>F+7$$\"3[LL$e9ui2%F-F/7$$\"3nmmm\"z_\" 4iF-F27$$\"3[mmmT&phN)F-F57$$\"3BLLe*=)H\\5!#=F87$$\"3gmm\"z/3uC\"F:F< 7$$\"3%)***\\7LRDX\"F:F?7$$\"3^mm\"zR'ok;F:FB7$$\"3w***\\i5`h(=F:FE7$$ \"3YLLL3En$4#F:FH7$$\"3qmm;/RE&G#F:FK7$$\"3\")*****\\K]4]#F:FN7$$\"3$* *****\\PAvr#F:FQ7$$\"3(******\\nHi#HF:FT7$$\"3jmm\"z*ev:JF:FW7$$\"3?LL L347TLF:FZ7$$\"3,LLLLY.KNF:Fgn7$$\"3v***\\7o7Tv$F:Fjn7$$\"3'GLLLQ*o]RF :F]o7$$\"3A++D\"=lj;%F:F`o7$$\"31++vV&RY2aF:Fbp7$$\"39mm;zXu9cF:Fep7$$\"3l******\\y))GeF:Fhp7$$\"3' *)***\\i_QQgF:F[q7$$\"3@***\\7y%3TiF:F^q7$$\"36****\\P![hY'F:Faq7$$\"3 jKLL$Qx$omF:Fdq7$$\"3!)*****\\P+V)oF:Fgq7$$\"3?mm\"zpe*zqF:Fjq7$$\"3%) *****\\#\\'QH(F:F]r7$$\"3GKLe9S8&\\(F:F`r7$$\"3R***\\i?=bq(F:Fcr7$$\"3 \"HLL$3s?6zF:Ffr7$$\"3a***\\7`Wl7)F:Fir7$$\"3#pmmm'*RRL)F:F\\s7$$\"3Qm m;a<.Y&)F:F_s7$$\"3Bs%F:7$FQ$\"3JJ.(*3rDDWF:7$FT$\"3@6Khqznc TF:7$FW$\"3eB]om9$p#RF:7$FZ$\"3#pQCux4-n$F:7$Fgn$\"3E-.x#>3fY$F:7$Fjn$ \"3JW0Fn:_UKF:7$F]o$\"37P!49`Ho0$F:7$F`o$\"3Wk]P6rIlGF:7$Fco$\"31nv#\\ R(4%p#F:7$Ffo$\"3j!G$)RBaj_#F:7$Fio$\"3V]o!)R)G:Q#F:7$F\\p$\"3(QLO143Y B#F:7$F_p$\"3K'HNR(ze\"4#F:7$Fbp$\"3>3(oO$=cu>F:7$Fep$\"3\\R^b=F:7 $Fhp$\"3'HHz!3&e+u\"F:7$F[q$\"31AvIRH1M;F:7$F^q$\"3)zRTvtcw`\"F:7$Faq$ \"3P;4N-JEP9F:7$Fdq$\"3_L8f\\%eEN\"F:7$Fgq$\"3vR+LV]\"yE\"F:7$Fjq$\"39 +F5B%Rb>\"F:7$F]r$\"3q,#zeBH77\"F:7$F`r$\"3R0ZY#)>`b5F:7$Fcr$\"3T%)4CG kq4**F-7$Ffr$\"3U!3]`m&p;$*F-7$Fir$\"38nFKoZ&Qt)F-7$F\\s$\"3,7fULh+2#) F-7$F_s$\"3*Ru%eB13,xF-7$Fbs$\"3w[hV4::IsF-7$Fes$\"3d71#zM[G#oF-7$Fhs$ \"3!z**fsNtTQ'F-7$F[t$\"3x)*['3+nd,'F-7$F^t$\"37cJe,gNYcF-7$Fat$\"3=XT r,![SJ&F-7$Fdt$\"3W%R'yOoqy\\F--Fgt6&FitF(FjtF(-%+AXESLABELSG6$Q\"x6\" Q!Fg^l-%%VIEWG6$;F(Fdt%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fsolve(g(x)=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+ " 0 "" {MPLTEXT 1 0 39 "v:=n->if n=0 then 1. else g(v(n-1)) fi:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(v(k),k=0..20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "67$\"\"\" \"\"!$\"+Poqy\\!#6$\"+z'zDh)!#5$\"+8z&)[vF($\"+5mYtzF+$\"+T^HW\"*F($\" +!y?3g(F+$\"+S-fA5F+$\"+jh9etF+$\"+'=9)*4\"F+$\"+/\"Q'*=(F+$\"+\\<%o: \"F+$\"+X?onqF+$\"+32&**>\"F+$\"+Jk'o(pF+$\"+/C4L7F+$\"+zD%y!pF+$\"+TC *)e7F+$\"+&=#eaoF+$\"+I(o\"z7F+$\"+'48I\"oF+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 412 "Les suites extraites d'indices pairs et impairs semble nt respectivement croissante et d\351croissante (on le prouverait comm e pour u) et localis\351es dans [0,1], d'o\371 les convergences. Par \+ contre, il semblerait que leurs limites l1 et l2 soient diff\351rentes . Elles sont solutions de l'\351quation g(g(x))=x. Ici encore, l'uniqu e X tel que g(X)=X v\351rifie \351galement g(g(X))=X, mais on va voir \+ qu'il y a d'autres solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(\{g(g(x)),x\},x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 345 169 169 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3dmmm;arz@!#>F+7 $$\"3[LL$e9ui2%F-F/7$$\"3nmmm\"z_\"4iF-F27$$\"3[mmmT&phN)F-F57$$\"3BLL e*=)H\\5!#=F87$$\"3gmm\"z/3uC\"F:F<7$$\"3%)***\\7LRDX\"F:F?7$$\"3^mm\" zR'ok;F:FB7$$\"3w***\\i5`h(=F:FE7$$\"3YLLL3En$4#F:FH7$$\"3qmm;/RE&G#F: FK7$$\"3\")*****\\K]4]#F:FN7$$\"3$******\\PAvr#F:FQ7$$\"3(******\\nHi# HF:FT7$$\"3jmm\"z*ev:JF:FW7$$\"3?LLL347TLF:FZ7$$\"3,LLLLY.KNF:Fgn7$$\" 3v***\\7o7Tv$F:Fjn7$$\"3'GLLLQ*o]RF:F]o7$$\"3A++D\"=lj;%F:F`o7$$\"31++ vV&RY2aF:Fbp7$$\"39mm;zXu9cF: Fep7$$\"3l******\\y))GeF:Fhp7$$\"3'*)***\\i_QQgF:F[q7$$\"3@***\\7y%3Ti F:F^q7$$\"36****\\P![hY'F:Faq7$$\"3jKLL$Qx$omF:Fdq7$$\"3!)*****\\P+V)o F:Fgq7$$\"3?mm\"zpe*zqF:Fjq7$$\"3%)*****\\#\\'QH(F:F]r7$$\"3GKLe9S8&\\ (F:F`r7$$\"3R***\\i?=bq(F:Fcr7$$\"3\"HLL$3s?6zF:Ffr7$$\"3a***\\7`Wl7)F :Fir7$$\"3#pmmm'*RRL)F:F\\s7$$\"3Qmm;a<.Y&)F:F_s7$$\"3gF-7$F/$ \"31EMcJ[8KqF-7$F2$\"3CgRzm8')*G)F-7$F5$\"3\"=Fm*pF'Ho*F-7$F8$\"3]L.e; /O>6F:7$F<$\"3eeVI-D7q7F:7$F?$\"3$y;F:7$FE$ \"3#)yA'z,24\"=F:7$FH$\"3C?([?/Gt,#F:7$FK$\"3m8JZg+-1AF:7$FN$\"3g&>Y^K S^U#F:7$FQ$\"39l2;:$)=^EF:7$FT$\"3&eFYIFYO(GF:7$FW$\"3b6ABHXpyIF:7$FZ$ \"3]DdR>X6ZLB%F:7$Fco$\"3x:pT0^YcWF:7$Ffo$\"3ndU=*>mko%F :7$Fio$\"3;Lr*o1qX*[F:7$F\\p$\"3/EkER#H^6&F:7$F_p$\"3-\\6*>^v$R`F:7$Fb p$\"3])=yX')e,`&F:7$Fep$\"3*=_SgCM7t&F:7$Fhp$\"3!oQvy!GALfF:7$F[q$\"3_ ZV&4;B\\7'F:7$F^q$\"3+\\^fzQl/jF:7$Faq$\"3%[+RYTEu\\'F:7$Fdq$\"3X/G_c3 XkmF:7$Fgq$\"3?M=Y1&ei$oF:7$Fjq$\"3.)=toi0h)pF:7$F]r$\"3N1-$HA'fVrF:7$ F`r$\"37l\"*[\\sy&G(F:7$Fcr$\"3-J()fZnFGuF:7$Ffr$\"3izh%)*p5;c(F:7$Fir $\"3C9b-j)*)\\p(F:7$F\\s$\"38Ty_Y*yv\"yF:7$F_s$\"3iKsWWt8PzF:7$Fbs$\"3 \">wXL2o+0)F:7$Fes$\"3uqptOg.\\\")F:7$Fhs$\"3hQUTm\"))pD)F:7$F[t$\"3lU @`O@v[$)F:7$F^t$\"3QWB[].zT%)F:7$Fat$\"3*Hw,)y%pj_)F:7$Fdt$\"3*Hy[$z'z Dh)F:-Fgt6&FitF(FjtF(-%+AXESLABELSG6$Q\"x6\"Q!Fi^l-%%VIEWG6$;F(Fdt%(DE FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(g(g(x ))-x,x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 306 164 164 {PLOTDATA 2 "6 %-%'CURVESG6$7S7$$\"\"!F)$\"3W%R'yOoqy\\!#>7$$\"3dmmm;arz@F,$\"3xhaN4! p,%QF,7$$\"3[LL$e9ui2%F,$\"3f#4Idoge&HF,7$$\"3nmmm\"z_\"4iF,$\"3d$HF^d 323#F,7$$\"3[mmmT&phN)F,$\"3L0'*HGKzE8F,7$$\"3BLLe*=)H\\5!#=$\"3`E+q*p Ai+(!#?7$$\"3gmm\"z/3uC\"FD$\"3Z)>p(QaWrAFG7$$\"3%)***\\7LRDX\"FD$!3[, A)y2(G&f\"FG7$$\"3^mm\"zR'ok;FD$!3'y214fT>b%FG7$$\"3w***\\i5`h(=FD$!3 \"Q4s(G)3Y_'FG7$$\"3YLLL3En$4#FD$!3XA8YGmXMwFG7$$\"3qmm;/RE&G#FD$!3)GI b$pVQCzFG7$$\"3\")*****\\K]4]#FD$!34@/Q&)***4e(FG7$$\"3$******\\PAvr#F D$!3OzM#R)fSLmFG7$$\"3(******\\nHi#HFD$!3$HTs`>S$e_FG7$$\"3jmm\"z*ev:J FD$!3r3bWoo81PFG7$$\"3?LLL347TLFD$!3;p2w$*)QYf\"FG7$$\"3,LLLLY.KNFD$\" 3*y10A@OlJ$!#@7$$\"3v***\\7o7Tv$FD$\"3)3_Ogsx#GEFG7$$\"3'GLLLQ*o]RFD$ \"3P0lnpb#zi%FG7$$\"3A++D\"=lj;%FD$\"3!eB(fpI>)p'FG7$$\"31++vV&Rw6\"F,7$$\"3emm;/T1&*\\FD$\"3o%f(*4N^1?\"F,7$$\"3&em;zRQb@& FD$\"3oJ[uS6PQ7F,7$$\"3\\***\\(=>Y2aFD$\"3>!*=Ge%ppA\"F,7$$\"39mm;zXu9 cFD$\"3dd&Q(om*[;\"F,7$$\"3l******\\y))GeFD$\"3erQvy&\\L/\"F,7$$\"3'*) ***\\i_QQgFD$\"33c[VX)*y`')FG7$$\"3@***\\7y%3TiFD$\"3#)y\\^M)4pN'FG7$$ \"36****\\P![hY'FD$\"3(>d+RrPy7$FG7$$\"3jKLL$Qx$omFD$!3'e=G0\"o_ERF^q7 $$\"3!)*****\\P+V)oFD$!33hl\"Q&o=/[FG7$$\"3?mm\"zpe*zqFD$!3d(4W*)3'F,7$$\"3WSsT \"*F,7$$\"3HmmmmxGp$*FD$!3kBX8Ic`?5FD7$$\"3B++D\"oK0e*FD$!3&eln2LU(Q6F D7$$\"3B++v=5s#y*FD$!3BP#[*R:Nc7FD7$$\"\"\"F)$!3-<7l?.U(Q\"FD-%'COLOUR G6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fe[l-%%VIEWG6$;F(Ffz%( DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "C'est plus clair, non ? \nIl semblerait qu'il n'y ait qu'un point fixe. Pour cela, on consid \350re la diff\351rence f(f(x))-x" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fsolve(g(g(x))=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+L))>h8!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(g( g(x))=x,x=.2..0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ " 0 "" {MPLTEXT 1 0 26 "fsolve(g(g(x))=x,x=.5.. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BwRZm!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 421 "Une \351tude de x->gog(x)-x etablirait effecti vement l'existence de trois solutions pour l'\351quation g(g(l))=l. Si on note x1,x2,x3 ces solutions (dans l'ordre croissant), on montrerai t que l'intervalle [0,x1] est stable par gog, donc tous les termes pai rs de v sont dans cet intervalle, donc major\351s par x1, donc leur li mite \351galement, donc cette limite EST x1. De meme, on montrerait qu e les termes impairs tendent vers x3." }}}}}}{MARK "2 5 3 1 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }