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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "TD 1 : Etudes de suites" 
}}{PARA 0 "" 0 "" {TEXT -1 24 "Jean-SŽbastien ROY, 1999" }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 46 "Ex 1 : Outils ŽlŽmentaires d'Žtude d'une \+
suite" }}{PARA 0 "" 0 "" {TEXT -1 42 "On consid�re la suite u(i)=1/i^2
, pour i>0" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "DŽfinition" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "u := i -> 1/i^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"uGR6#%\"iG6\"6$%)operatorG%&arrowGF(*$9$!\"#F(F(6\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Calcul des premiers termes" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(u(i),i=1..10);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6,\"\"\"#F#\"\"%#F#\"\"*#F#\"#;#F#\"#D#F#\"#O#
F#\"#\\#F#\"#k#F#\"#\")#F#\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
9 "Monotonie" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "d:=simplify
(u(i)-u(i-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG,$*(,&%\"iG\"
\"#!\"\"\"\"\"F+F(!\"#,&F(F+F*F+F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 86 "A noter : assume / is ne marchent quasiment JAMAIS. Ici, c'est \+
juste pour le principe." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
assume(i>=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "is(d<0);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "On efface l'assume" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "i:='i';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 67 "Trace l'evolution de la suite (bien noter
 la position des crochets)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "plot([seq([i,u(i)],i=1..10)],style=point);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 303 212 212 {PLOTDATA 2 "6$-%'CURVESG6$7,7$$\"\"\"\"\"!F(7$$
\"\"#F*$\"1+++++++D!#;7$$\"\"$F*$\"166666666F07$$\"\"%F*$\"1++++++]i!#
<7$$\"\"&F*$\"1+++++++SF;7$$\"\"'F*$\"1yxxxxxxFF;7$$\"\"(F*$\"171`Ej\"
3/#F;7$$\"\")F*$\"1+++++]i:F;7$$\"\"*F*$\"1oXB,zcM7F;7$$\"#5F*$\"1++++
+++5F;-%'COLOURG6&%$RGBG$FW!\"\"F*F*-%&STYLEG6#%&POINTG" 1 5 0 1 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "Limite de la suite" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "limit(u(i),i=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "Ex 2 : Convergence d'une
 sŽrie" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 " Tout d'abord, on dŽfinit une fonc
tion renvoyant le terme gŽnŽral du DSE de ln(1+x) :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "terme := (i,x) -> (-1)^(i+1)*x^i/i;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&termeGR6$%\"iG%\"xG6\"6$%)operatorG
%&arrowGF)*&*&)!\"\",&9$\"\"\"F3F3F3)9%F2F3\"\"\"F2!\"\"F)F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Puis on dŽfini une fonction renvoy
ant la somme des " }{TEXT 256 1 "n" }{TEXT -1 20 " premiers du DSE en \+
" }{TEXT 257 1 "x" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "approxLn := proc(n,x) local i;add(evalf(terme(i,x)),i
=1..n);end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)approxLnGR6$%\"nG%\"
xG6#%\"iG6\"F+-%$addG6$-%&evalfG6#-%&termeG6$8$9%/F5;\"\"\"9$F+F+F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "A noter : on a utilisŽ " }{TEXT 
258 3 "add" }{TEXT -1 12 " et non pas " }{TEXT 259 3 "sum" }{TEXT -1 
118 ". Le premier effectue la somme numŽrique terme a terme, alors que
 le second est adaptŽ aux calculs symboliques (quand " }{TEXT 263 1 "n
" }{TEXT -1 30 " n'est pas connu par exemple)." }}{PARA 0 "" 0 "" 
{TEXT -1 24 "De plus, on effectue un " }{TEXT 260 5 "evalf" }{TEXT -1 
95 " sur le terme ˆ ajouter, afin de ne jamais conserver de fractions,
 qui ralentiraient le calcul." }}{PARA 0 "" 0 "" {TEXT -1 44 "Enfin, l
'utilisation d'une variable locale (" }{TEXT 261 5 "local" }{TEXT -1 
46 ") n'est pas obligatoire, mais prŽfŽrable avec " }{TEXT 262 3 "add
" }{TEXT -1 20 " (raisons obscures)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 44 "L'approximation de ln(2) fournie par Maple :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(ln(2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+1=ZJp!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Pu
is la notre avec 10000 termes :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "approxLn(10000,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P=(4
$p!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Utiliser le DSE de ln((
1+x)/(1-x)) revient a utiliser le DSE de ln(1+x) et ln(1-x) en x tq (1
+x)/(1-x)=2" }}{PARA 0 "" 0 "" {TEXT -1 11 "On resoud :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve((1+x)/(1-x)=2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 42 "On applique notre formule avec 17 termes :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "approxLn(17,1/3)-approxLn(17,-1/3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/=ZJp!#5" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 159 "Le resultat est bien meilleur. L'explication tient dan
s le terme en x^i du DSE. Quand x=1/3, il tend tr�s rapidement vers 0,
 ce qui n'est pas le cas quand x=1." }}}}{SECT 0 {PARA 256 "" 0 "" 
{TEXT -1 26 "Ex 3 : RŽcurrence linŽaire" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Defin
ition d'une suite recurrente :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
208 "'option remember' rends le calcul beaucoup plus rapide, en memori
sant toutes les valeurs calculees de la suite, ce qui, pour une suite \+
doublement rŽcursive, Žvite de calculer de nombreuses fois chaque vale
ur." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Attention ˆ l'ordre et ˆ \+
la syntaxe ; d'abord f:=proc ... et ensuite la definition des valeurs \+
initiales." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f:=proc(n) op
tion remember; 10*f(n-1)-16*f(n-2); end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "f(1):=20:f(2):=-8:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 26 "Calcul des premiers termes" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "seq(f(i),i=1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'
\"#?!\")!$+%!%sQ!&?B$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Resolut
ion de la recurence (utiliser un nom different pour la suite (ici u ˆ \+
la place de f), sinon maple cherche a calculer sa valeur symboliquemen
t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "rsolve(\{u(n)=10*u(n-
1)-16*u(n-2),u(1)=20,u(2)=-8\},u(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,&)\"\")%\"nG!\"\")\"\"#F&\"#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
71 "Definition non recursive (ne pas mettre de -> quand on utilise una
pply)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "u:= unapply(%,n);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"nG6\"6$%)operatorG%&ar
rowGF(,&)\"\")9$!\"\")\"\"#F/\"#9F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 19 "Calcul de la limite" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "limit(u(n),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Calcu
l d'un equivalent en +infinity (ne pas oublier de mettre un ordre)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "asympt(u(n),n,2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&)\"\")%\"nG!\"\"-%\"OG6#)\"\"#F&\"\"\"" }}
}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 52 "Ex 4.1 : RŽcurence du type u(n+
1)=f(u(n)), exemple 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest
art;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:= x -> sqrt(3*x-2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&
arrowGF(-%%sqrtG6#,&9$\"\"$!\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 93 "Ici 'option remember' n'est pas vraiment necessaire (la
 suite n'est pas doublement recursive)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "u:= n -> f(u(n-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"uGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%\"fG6#-F$6#,&9$\"\"\"!\"\"
F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "u(0):=1.5;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6#\"\"!$\"#:!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "u(3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+LC2B<!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(
u(i),i=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,$\"+I)Q6e\"!\"*$\"+.
iKc;F%$\"+LC2B<F%$\"+k&H-y\"F%$\"+'4bx#=F%$\"+%z]j'=F%$\"+go6(*=F%$\"+
\"y)G@>F%$\"+W'o+%>F%$\"+5Z`a>F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
70 "La fonction marche renvoie les extremites d'un petit segment verti
cal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "marche:= i -> ([u(i
),u(i)],[u(i),u(i+1)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "En tr
acant plusieurs marche on cree l'escalier. (bien noter la position des
 crochets, comme toujours)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "seq(marche(i),i=0..2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6(7$$\"#:!
\"\"F$7$F$$\"+I)Q6e\"!\"*7$F(F(7$F($\"+.iKc;F*7$F-F-7$F-$\"+LC2B<F*" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot([%]);" }}{PARA 13 "" 
1 "" {GLPLOT2D 215 147 147 {PLOTDATA 2 "6%-%'CURVESG6$7(7$$\"1+++++++:
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "En nomant chacun des graphes (ave
c un ':' a la fin des expressions pour eviter l'horrible charabia de m
aple). Et en utilisant plots[display] on arrive a superposer des graph
es." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p1:=plot([seq(marche
(i),i=0..20)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Avec [f(x),x] \+
on trace simultanement y=f(x) et y=x." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "p2:=plot([f(x),x],x=0.5..2.5,color=[green,blue]):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plots[display](\{p1,p2\},sc
aling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 335 214 214 
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f\">F*F`fl7$Ff[lFf[l7$$\"1mm\"H!o-**>F*Fdfl7$$\"1++DTO5T?F*Fgfl7$$\"1n
mmT9C#3#F*Fjfl7$$\"1++D1*3`7#F*F]gl7$$\"1LLL$*zym@F*F`gl7$$\"1LL$3N1#4
AF*Fcgl7$$\"1nm\"HYt7D#F*Ffgl7$F^^lF^^l7$$\"1mm;9@BMBF*Fjgl7$$\"1LLL`v
&QP#F*F]hl7$$\"1++DOl5;CF*F`hl7$$\"1++v.UacCF*Fchl7$Fg_lFg_l-Fip6&F[qF
_qF_qF]`l-%+AXESLABELSG6$%!GF[il-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;
$\"\"&F^q$\"#DF^q%(DEFAULTG" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Les points fixes :
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{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 53 "D(f) represente la fonction derivee de la fonction f." 
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{MPLTEXT 1 0 15 "evalf(D(f)(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+++++v!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Le point x=2 est a
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;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++:!\"*" }}}{EXCHG {PARA 0 
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{TEXT -1 54 "On choisit une nouvelle valeur initiale pour la suite." }
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{PARA 0 "" 0 "" {TEXT -1 57 "A partir de quelques termes, la suite n'e
st plus dŽfinie." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 50 "Ex 4.2 : RŽ
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103 "Pour le corrigŽ de cet exercice, on se rŽf�rera au corrigŽ du TD \+
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{TEXT -1 42 "Ex 4.3 : RŽcurrence du type u(n+1)=f(u(n))" }}{EXCHG 
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$FcalFcal7$Fcal$\"1+++vf>17Fhu7$FgalFgal7$Fgal$\"1+++lVXF7Fhu7$F[blF[b
l7$F[bl$\"1+++4@K`7Fhu7$F_blF_bl7$F_bl$\"1+++b\"3aG\"Fhu7$FcblFcbl7$Fc
bl$\"1+++jq8E8Fhu7$FgblFgbl7$Fgbl$\"1+++b(>$z8Fhu7$F[clF[cl7$F[cl$\"1+
++%\\h7X\"Fhu7$F_clF_cl7$F_cl$\"1+++i*zIb\"Fhu7$FcclFccl7$Fccl$\"1+++%
oGgq\"Fhu7$FgclFgcl7$Fgcl$\"1+++OpEb>Fhu7$F[dlF[dl7$F[dl$\"1+++'RM:T#F
hu-F\\[l6&F^[l$\"#5F)F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%(SCALINGG6#%,CONS
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45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "La suit
e est croissante, elle ne peut etre majoree car sinon elle convergerai
t vers une limite superieure a 1 (ce qui est impossible)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 12 "Autres cas :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "assume(x<-1):is(f(x)>1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:
='x':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Donc si u(0)< -1, u(1)>1
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{TEXT -1 72 "Si u(0)=1 ou u(0)=-1, la suite est constante egale a 1 a \+
partir de u(1)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Si 0<u(0)<1 :
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "assume(x>-1,x<1):is(f(x
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6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x':" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Donc pour tout n > 1 , u(n) est c
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0 "> " 0 "" {MPLTEXT 1 0 50 "p4:=plot([f(x),x],x=-0.4..1.2,color=[gree
n,blue]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plots[display]
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l$\"#7Fedl%(DEFAULTG" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 
45.000000 0 }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 39 "Ex 5 : Recurence
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start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq1:=n*u(n+2)-5*u
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\"\"\"-%\"uG6#,&F'F(\"\"#F(F(F(-F*6#,&F'F(F(F(!\"&*&F0F(-F*6#F'F(!\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "L'operateur '.' concatene les
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a . 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#a2G" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "add((a.i)*n^i,i=0..3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*%#a0G\"\"\"*&%#a1GF%%\"nGF%F%*&%#a2GF%F(\"\"#F%*&%#a3
GF%F(\"\"$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p:=unapply(
%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGR6#%\"nG6\"6$%)operator
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\"$F.F(F(6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Ne pas se tromper
 sur l'ordre des parametres de la fonction subs." }}{PARA 0 "" 0 "" 
{TEXT -1 52 "Ici on substitue une fonction (u) par une autre (p)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(u=p,eq1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*&%\"nG\"\"\"-%\"pG6#,&F%F&\"\"#F&F&F&-F(6#,&F
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1 0 13 "collect(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&,&%#a2G!
\"#%#a3G!\"$\"\"\"%\"nG\"\"#F**&,(%#a1G!\"%F(!\"(F&!\"'F*F+F*F*%#a0GF2
F(!\"&F&F4F/F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Attention. si \+
on avait divisŽ eq1 par n, coeffs ne marcherait pas car l'expresion ne
 serait pas vraiment un polynome." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "coeffs(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,(%#a
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(\{%\});" }}{PARA 11 "
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{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#a1G\"\"\"%\"nGF&F&*&F%F&F'\"\"#!
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1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 
"Puis la convergence uniforme sur [a,1], a>0" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "simplify(l(x)-f(x,n));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&*&,&*$)%\"xG\"\"#\"\"\"\"\"\"F+F+F+-%$expG6#,$F(!\"\"
F+F*,&*&%\"nGF+F(F+F+F+F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 
"Pas d'espoir de maximiser sur un intervale de type [a,1] : maple ne l
'authorise pas." }}{PARA 0 "" 0 "" {TEXT -1 95 "On se rabbat sur la ma
ximisation du numerateur seulement (et on fait le denominateur a la ma
in)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "maximize(numer(%),x,
0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 52 "Donc l(x)-f(x,n)<1/(na+1) et la convergence unifor
me" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Non-convergence uniforme su
r ]0,1], par exemple avec x=1/n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "limit(l(1/n)-f(1/n,n),n=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "D'
ou le resultat souhaite." }}}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK 
"0 3 11 0 0" 82 }{VIEWOPTS 1 1 0 1 1 1803 }