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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "TD 1 : Etudes de suites (
CorrigŽ)" }}{PARA 0 "" 0 "" {TEXT -1 34 "Jean-SŽbastien ROY, Septembre
 1998" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Ex 1 : Les poissons" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Avant tout travail, toujours bien \+
penser ˆ faire un " }{TEXT 256 7 "restart" }{TEXT -1 26 ". Idem si rie
n ne va plus." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "On dŽfinit la fonction qui ˆ " }
{TEXT 315 4 "u(n)" }{TEXT -1 9 " associe " }{TEXT 316 6 "u(n+1)" }
{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x ->
 R*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operator
G%&arrowGF(*&%\"RG\"\"\"9$F.F(F(6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 42 "Puis la dŽfinition rŽcursive de la suite :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "u := n -> f(u(n-1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"uGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%\"fG6#-F$6#,&
9$\"\"\"!\"\"F3F(F(6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "La vale
ur initiale de la suite :" }}{PARA 0 "" 0 "" {TEXT -1 131 "(on la choi
sit flottante (i.e. ˆ virgule) pour que Maple n'effectue pas les calcu
ls exacts, ce qui prendrait un temps considŽrable)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "u(0):=2.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>-%\"uG6#\"\"!\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Et la v
aleur du param�tre " }{TEXT 317 1 "R" }{TEXT -1 2 " :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R:=1.5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RG$\"#:!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
28 "Calcul des premiers termes :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "seq(u(i),i=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
,$\"#I!\"\"$\"$]%!\"#$\"%]n!\"$$\"']75!\"%$\"(](=:!\"&$\")]7yA!\"'$\"*
](=<M!\"($\"+]7yD^!\")$\"+v=n)o(F:$\"+\"y+L:\"F7" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 70 "TracŽ de l'evolution de la suite (bien noter la po
sition des crochets)" }}{PARA 0 "" 0 "" {TEXT -1 33 "On trace la sŽque
nce des couples " }{TEXT 318 8 "[i,u(i)]" }{TEXT -1 2 " :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([seq([i,u(i)],i=1..10)],style=
point);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 212 212 {PLOTDATA 2 "6$-%'CU
RVESG6$7,7$$\"\"\"\"\"!$\"\"$F*7$$\"\"#F*$\"1+++++++X!#:7$F+$\"1++++++
]nF27$$\"\"%F*$\"1+++++]75!#97$$\"\"&F*$\"1+++++v=:F;7$$\"\"'F*$\"1+++
+]7yAF;7$$\"\"(F*$\"1++++v=<MF;7$$\"\")F*$\"1+++]7yD^F;7$$\"\"*F*$\"1+
++v=n)o(F;7$$\"#5F*$\"1+++\"y+L:\"!#8-%'COLOURG6&%$RGBG$FW!\"\"F*F*-%&
STYLEG6#%&POINTG" 1 5 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Expression non-rŽcursive de la \+
suite :" }}{PARA 0 "" 0 "" {TEXT -1 143 "(L'exemple traitŽ ici est tri
vial. Maple sait traiter les recurences linŽaires ˆ coefficients const
ants et certains autres cas. Voir l'aide de " }{TEXT 258 6 "rsolve" }
{TEXT -1 4 " :  " }{TEXT 257 7 "?rsolve" }{TEXT -1 1 ")" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 65 "On veut rŽsoudre quelque soit R, donc on \+
efface sa valeur ainsi :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "
R:='R';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RGF$" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 64 "Puis on rŽsoud la rŽcurrence. Bien noter que l'
on n'utilise pas " }{TEXT 259 1 "u" }{TEXT -1 6 " mais " }{TEXT 260 1 
"v" }{TEXT -1 30 ", qui n'est pas encore dŽfini." }}{PARA 0 "" 0 "" 
{TEXT -1 45 "En effet, Maple chercherait sinon ˆ calculer " }{TEXT 
261 4 "u(n)" }{TEXT -1 5 ", et " }{TEXT 262 1 "n" }{TEXT -1 58 " n'Žta
nt pas dŽfini, la rŽcurrence ne s'arreterait jamais." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "e:=rsolve(v(n+1)=R*v(n),v(n));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"eG*&-%\"vG6#\"\"!\"\"\")%\"RG%\"nGF*" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "DŽmonstration que " }{TEXT 319 4 
"u(n)" }{TEXT -1 19 " ne convenait pas :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "u(n);" }}{PARA 8 "" 1 "" {TEXT -1 42 "Error, (in u) to
o many levels of recursion" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "On \+
aimerait maintenant dŽfinir la suite " }{TEXT 264 4 "v(n)" }{TEXT -1 
34 ", pour cela il faut transformer l'" }{TEXT 266 10 "expression" }
{TEXT -1 1 " " }{TEXT 265 1 "e" }{TEXT -1 8 " en une " }{TEXT 267 8 "f
onction" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "C'est l'objet \+
de la fonction " }{TEXT 263 28 "unapply(expression,variable)" }{TEXT 
-1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=unapply(e,n)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGR6#%\"nG6\"6$%)operatorG%&a
rrowGF(*&-F$6#\"\"!\"\"\")%\"RG9$F0F(F(6\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 11 "Comme pour " }{TEXT 268 4 "u(n)" }{TEXT -1 33 ", on dŽfin
it la valeur initiale :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v
(0):=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"vG6#\"\"!\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Et la valeur du parametre :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R:=1.5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RG$\"#:!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
10 "D�s lors :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$)$\"#:!\"\"%\"nG\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "Limite de la suite :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "limit(v(n),n=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%)infinityG" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 41 "E
x 2 : Trop de poissons nuit aux poissons" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "M�me principe :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:= x -> R
*x*(1-x/100000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6
$%)operatorG%&arrowGF(*(%\"RG\"\"\"9$F.,&F.F.F/#!\"\"\"'++5F.F(F(6\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u:= n -> f(u(n-1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"nG6\"6$%)operatorG%&arrowG
F(-%\"fG6#-F$6#,&9$\"\"\"!\"\"F3F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "R:=1.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG$\"#:
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "u(0):=2.0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6#\"\"!$\"#?!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "On aurait pu se passer de la fonction " }
{TEXT 269 1 "f" }{TEXT -1 36 ". Malheureusement, la dŽfinition de " }
{TEXT 270 4 "u(n)" }{TEXT -1 47 " aurait fait intervenir deux fois le \+
calcul de " }{TEXT 272 6 "u(n-1)" }{TEXT -1 22 ". Ainsi pour calculer \+
" }{TEXT 273 5 "u(10)" }{TEXT -1 39 ", Maple aurait eu ˆ calculer deux
 fois " }{TEXT 271 3 "u(9" }{TEXT -1 15 "), quatre fois " }{TEXT 274 
4 "u(8)" }{TEXT -1 12 ", huit fois " }{TEXT 275 4 "u(7)" }{TEXT -1 
105 "... et le temps de calcul aurait rapidement ŽtŽ prohibitif. Avec \+
la dŽfinition prŽcŽdente, pour calculer " }{TEXT 276 4 "u(n)" }{TEXT 
-1 25 ", Maple calcule une fois " }{TEXT 277 6 "u(n-1)" }{TEXT -1 14 "
, et applique " }{TEXT 278 1 "f" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 66 "On verra plus loin une autre mŽthode pour circonveni
r au probl�me." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(u(i),
i=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$\"+++%***H!\"*$\"+0]x*\\
%F%$\"+)ye$\\nF%$\"+([NB,\"!\")$\"+3&\\$=:F,$\"+a%yrF#F,$\"+])*)\\T$F,
$\"+_at?^F,$\"+*))prn(F,$\"+*R\"p]6!\"(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "Tra�ons l'escalier :" }}{PARA 0 "" 0 "" {TEXT -1 12 "La f
onction " }{TEXT 320 6 "marche" }{TEXT -1 96 " renvoie les extremitŽs \+
d'un petit segment vertical sous forme d'une sŽquence de deux couples \+
: " }{TEXT 280 11 "[u(i),u(i)]" }{TEXT -1 12 " (dŽbut) et " }{TEXT 
279 13 "[u(i),u(i+1)]" }{TEXT -1 7 " (fin)." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 tra�ant plusieurs marche on cr
Že 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$$\"#?!\"\"F$7$F$$\"+++%***H!\"*7$F(
F(7$F($\"+0]x*\\%F*7$F-F-7$F-$\"+)ye$\\nF*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "Les traits horizontaux apparaissent car Maple fait un tra
it entre chaque point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "p
lot([%]);" }}{PARA 13 "" 1 "" {GLPLOT2D 215 147 147 {PLOTDATA 2 "6#-%'
CURVESG6$7(7$$\"\"#\"\"!F(7$F($\"1+++++%***H!#:7$F,F,7$F,$\"1+++0]x*\\
%F.7$F1F17$F1$\"1+++)ye$\\nF.-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*" 1 2 0 
1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 40 "En nommant chacun des graphes (avec un '" }{TEXT 321 1 
":" }{TEXT -1 86 "' a la fin des expressions pour eviter l'horrible ch
arabia de maple), et en utilisant " }{TEXT 322 14 "plots[display]" }
{TEXT -1 36 " on arrive ˆ superposer des graphes." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "p1:=plot([seq(marche(i),i=0..40)]):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Avec [f(x),x] on trace simultanŽme
nt y=f(x) et y=x." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p2:=pl
ot([f(x),x],x=0..100000,color=[green,blue]):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 9 "L'option " }{TEXT 292 19 "scaling=constrained" }{TEXT 
-1 29 " donne des axes orthonormŽs :" }}}{EXCHG {PARA 0 "> " 0 "" 
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{TEXT -1 63 "La suite converge encore. Essayons une valeur plus ŽlevŽe
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{TEXT -1 68 "La suite ne converge plus. En fait elle oscille entre deu
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Pourquoi le compor
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53 "Pour cela il faut Žtudier le comportement des suites " }{TEXT 289 
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e qui correspond ˆ Žtudier " }{TEXT 291 5 "f o f" }{TEXT -1 1 "." }}}
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{TEXT 297 4 "f@@2" }{TEXT -1 10 " signifie " }{TEXT 296 5 "f o f" }
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(-Fe[l6&Fg[lFh[lF(F(" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "La population des \+
poissons cro”t tr�s rapidement (" }{TEXT 302 5 "R=2.5" }{TEXT -1 204 "
). Malheureusement, ceci finit par faire cro”tre le nombre de godzilla
s de mani�re tr�s importante, ceux-ci mangent tous les poissons, puis \+
s'Žteignent. Il suffit de peu de choses pour rompre l'Žquilibre." }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "Ex 4 : Suite de fonctions" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 35 "Etudions la convergence simple de :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f:= (x,n) -> n*cos(x)^n*sin(x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"nG6\"6$%)operatorG%&a
rrowGF)*(9%\"\"\")-%$cosG6#9$F.F/-%$sinGF3F/F)F)6\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 12 "Sur [0,Pi/2]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 1 "(" }{TEXT 305 6 "assume" }{TEXT -1 49 " permet d'imposer une con
trainte ˆ une variable. " }{TEXT 304 12 "additionally" }{TEXT -1 67 " \+
permet d'ajouter une nouvelle contrainte sans effacer la premi�re)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "assume(x>0);additionally(x<
Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit(f(x,n),n=in
finity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 64 "La suite de fonction converge simplement vers la f
onction nulle." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "TracŽ des graph
es :" }}{PARA 0 "" 0 "" {TEXT -1 1 "(" }{TEXT 306 6 "red$10" }{TEXT 
-1 12 " equivaut ˆ " }{TEXT 307 15 "red,red,...,red" }{TEXT -1 12 " (d
ix fois))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([seq(f(x,
n),n=1..10)],x=0..Pi/2,scaling=constrained,color=[red$10]);" }}{PARA 
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{TEXT -1 24 "Pour quelles valeurs de " }{TEXT 310 1 "x" }{TEXT -1 2 ",
 " }{TEXT 311 1 "g" }{TEXT -1 16 ", la dŽrivŽe de " }{TEXT 309 1 "f" }
{TEXT -1 19 ", s'annule-t'elle ?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "s:=[solve(g(x,n)=0,x)];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"sG7$-%'arctanG6#*$%\"nG#!\"\"\"\"#,$F&F," }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "x0:=unapply(s[1],n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#x0GR6#%\"nG6\"6$%)operatorG%&arrowGF(-%'arctanG6#*$9
$#!\"\"\"\"#F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(0,n
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"nG" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 7 "f(0,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
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1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Ainsi
, " }{TEXT 328 1 "g" }{TEXT -1 22 " s'annule une fois en " }{TEXT 329 
2 "x0" }{TEXT -1 7 " entre " }{TEXT 330 1 "0" }{TEXT -1 4 " et " }
{TEXT 331 4 "Pi/2" }{TEXT -1 20 ", est positive avant" }{TEXT 332 1 " \+
" }{TEXT 333 4 "x0(n" }{TEXT -1 8 ") (donc " }{TEXT 334 1 "f" }{TEXT 
-1 22 " est croissante entre " }{TEXT 335 1 "0" }{TEXT -1 4 " et " }
{TEXT 336 5 "x0(n)" }{TEXT -1 12 "), et comme " }{TEXT 337 14 "f(0)=f(
Pi/2)=0" }{TEXT -1 5 ", et " }{TEXT 338 1 "g" }{TEXT -1 27 " est de si
gne constant sur " }{TEXT 339 11 "x0(n)..Pi/2" }{TEXT -1 2 ", " }
{TEXT 340 1 "f" }{TEXT -1 37 " est nŽcessairement dŽcroissante sur " }
{TEXT 341 11 "x0(n)..Pi/2" }{TEXT -1 43 ". (toutes les fonctions manip
ulŽes sont C1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 9 "Etudions " }{TEXT 342 2 "x0" }{TEXT -1 2 " :" }}}{EXCHG 
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{XPPMATH 20 "6#,$*&%\"nG#!\"$\"\"#,&*$F%!\"\"\"\"\"F,F,F+#F+F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "... manifestement nŽgatif, donc " 
}{TEXT 343 2 "x0" }{TEXT -1 22 " est dŽcroissante et :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(x0(n),n=infinity);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "Donc, ˆ partir d'une certain rang, " }{TEXT 344 4 "x0<a" }{TEXT 
-1 4 " et " }{TEXT 345 1 "f" }{TEXT -1 20 " atteint son max en " }
{TEXT 346 1 "a" }{TEXT -1 17 ", qui tends vers " }{TEXT 347 1 "0" }
{TEXT -1 38 " (convergence simple, voir plus haut)." }}{PARA 0 "" 0 "
" {TEXT -1 5 "Donc " }{TEXT 348 1 "f" }{TEXT -1 9 " CU vers " }{TEXT 
349 1 "0" }{TEXT -1 5 " sur " }{TEXT 350 8 "[a,Pi/2]" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Converg
ence uniforme sur " }{TEXT 351 8 "[0,Pi/2]" }{TEXT 352 1 " " }{TEXT 
-1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 4 "Sur " }{TEXT 353 8 "[0,Pi/2]" }
{TEXT -1 23 ", f atteint son max en " }{TEXT 354 2 "x0" }{TEXT -1 2 " \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f(x0(n),n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*(%\"nG#\"\"\"\"\"#)*$,&*$F$!\"\"F&F&F&#F,F'
F$F&F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(%,n=infin
ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 33 "D'o� la non convergence uniforme." }}{PARA 0 "
" 0 "" {TEXT -1 23 "Calcul des intŽgrales :" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 7 "x:='x':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "int(f(x,n),x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limit
G6%,&*()-%$cosG6#%\"xG,&\"\"\"F.%\"nGF.F.F-!\"\"F/F.F0*&F-F0F/F.F./F,,
$%#PiG#F.\"\"#%%leftG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Pas de c
hance, Maple n'arrive pas ˆ calculer cette limite pourtant fort simple
." }}{PARA 0 "" 0 "" {TEXT -1 5 "Avec " }{TEXT 312 2 "op" }{TEXT -1 
40 " on extrait un morceau de l'expression :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "op(1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*()
-%$cosG6#%\"xG,&\"\"\"F+%\"nGF+F+F*!\"\"F,F+F-*&F*F-F,F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(x=Pi/2,%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*()-%$cosG6#,$%#PiG#\"\"\"\"\"#,&F,F,%\"nGF,F,F.!\"
\"F/F,F0*&F.F0F/F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sim
plify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%%\"nGF%!\"\"F
&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 109 "Manifestement diffŽrent de l'intŽgrale de la limite si
mple (0) : on ne peut pas Žchanger limite et intŽgrale." }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 40 "Ex 5 : RŽcurrence ˆ solution polynomiale
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 45 "On dŽfinit l'Žquation vŽrifiŽe par la sui
te :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq1:=n*u(n+2)-5*u(n
+1)-(n+1)*u(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G,(*&%\"nG\"
\"\"-%\"uG6#,&F'F(\"\"#F(F(F(-F*6#,&F'F(F(F(!\"&*&F0F(-F*6#F'F(!\"\"" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Puis un polyn™me quelconque de \+
degrŽ trois :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "(L'opŽrateur '" 
}{TEXT 303 1 "." }{TEXT -1 39 "' concat�ne les ŽlŽments autour de lui)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a . 2;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%#a2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "
add((a.i)*n^i,i=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%#a0G\"\"
\"*&%#a1GF%%\"nGF%F%*&%#a2GF%F(\"\"#F%*&%#a3GF%F(\"\"$F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p:=unapply(%,n);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"pGR6#%\"nG6\"6$%)operatorG%&arrowGF(,*%#a0G\"\"\"
*&%#a1GF.9$F.F.*&%#a2GF.F1\"\"#F.*&%#a3GF.F1\"\"$F.F(F(6\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "Ne pas se tromper sur l'ordre des param�t
res de la fonction " }{TEXT 355 4 "subs" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 31 "Ici on substitue une fonction (" }{TEXT 357 1 "u" }
{TEXT -1 17 ") par une autre (" }{TEXT 356 1 "p" }{TEXT -1 2 ")." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(u=p,eq1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*&%\"nG\"\"\"-%\"pG6#,&F%F&\"\"#F&F&F&-F(6#,&F
%F&F&F&!\"&*&F.F&-F(6#F%F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
26 "On regroupe les termes en " }{TEXT 358 9 "n^qqchose" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "collect(%,n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,.*&,&%#a2G!\"#%#a3G!\"$\"\"\"%\"nG\"\"#F**&,(%#a1G!\"%
F(!\"(F&!\"'F*F+F*F*%#a0GF2F(!\"&F&F4F/F4" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Attention : si on avait divisŽ " }{TEXT 359 3 "eq1" }
{TEXT -1 5 " par " }{TEXT 360 1 "n" }{TEXT -1 2 ", " }{TEXT 361 6 "coe
ffs" }{TEXT -1 70 " ne marcherait pas car l'expresion ne serait pas vr
aiment un polyn™me." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "On extrait
 les coefficients :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "coef
fs(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,(%#a1G!\"%%#a3G!\"(%#a2G!
\"',*%#a0GF)F&!\"&F(F,F$F,,&F(!\"#F&!\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 55 "Qui doivent tous �tre nuls si l'Žquation est vŽrifiŽe :" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(\{%\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<&/%#a3G,$%#a1G\"\"#/%#a0G\"\"!/%#a2G,$F'!\"
$/F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "D'o� l'ensemble des pol
yn™mes possibles :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(
%,p(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#a1G\"\"\"%\"nGF&F&*&
F%F&F'\"\"#!\"$*&F%F&F'\"\"$F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 
"VŽrification :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u:=unapp
ly(%,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"nG6\"6$%)opera
torG%&arrowGF(,(*&%#a1G\"\"\"9$F/F/*&F.F/F0\"\"#!\"$*&F.F/F0\"\"$F2F(F
(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(eq1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 29 "Ex 6 : DŽveloppements limitŽs" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 107 "On cherche un polyn™me unitaire quelconque de degrŽ 3 (m
anifestement nŽcessaire pour que la suite converge)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "P:=n*n*n+a*n*n+b*n+c;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"PG,**$%\"nG\"\"$\"\"\"*&%\"aGF)F'\"\"#F)*&%\"bGF)F'F)F)%\"cG
F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "L'expression de la suite :
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E:=P^(1/3)-(n^4+3*n^2)^
(1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG,&*$,**$%\"nG\"\"$\"\"
\"*&%\"aGF+F)\"\"#F+*&%\"bGF+F)F+F+%\"cGF+#F+F*F+*$,&*$F)\"\"%F+*$F)F.
F*#F+F6!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Que l'on dŽvelopp
e en sŽrie au voisinage de plus l'infini :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "S:=series(E,n=infinity,2);" }}{PARA 11 "" 0 "" 
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,(%\"aG#\"\"\"\"
\"$*&,(%\"bGF'*$F&\"\"##!\"\"\"\"*#!\"$\"\"%F(F(%\"nGF0F(-%\"OG6#*$F5!
\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "On supprime le O(1/n^2),
 qui converge absolument." }}{PARA 0 "" 0 "" {TEXT -1 24 "Le reste doi
t �tre nul :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "SP:=convert
(S,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SPG,&%\"aG#\"\"\"\"
\"$*&,(%\"bGF'*$F&\"\"##!\"\"\"\"*#!\"$\"\"%F(F(%\"nGF0F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F:=subs(n=1/m,SP);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"FG,&%\"aG#\"\"\"\"\"$*&,(%\"bGF'*$F&\"\"##!\"
\"\"\"*#!\"$\"\"%F(F(%\"mGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "collect(F,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG#\"\"
\"\"\"$*&,(%\"bGF%*$F$\"\"##!\"\"\"\"*#!\"$\"\"%F&F&%\"mGF&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "C:=\{coeffs(%,m)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG<$,$%\"aG#\"\"\"\"\"$,(%\"bGF(*$
F'\"\"##!\"\"\"\"*#!\"$\"\"%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "solve(C,\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"aG
\"\"!/%\"bG#\"\"*\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 6 "assign
" }{TEXT -1 58 " permet d'affecter aux variables les valeurs trouvŽes \+
par " }{TEXT 314 5 "solve" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "
Les polyn™mes convenant sont de la forme :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 2 "P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"nG\"\"$
\"\"\"F%#\"\"*\"\"%%\"cGF'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}
{MARK "0 3 26 1 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 }