{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 } {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 "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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 208 10 "Exercice 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 450 "On commence par remarquer que n\351cess airement le polyn\364me P doit \352tre de degr\351 3. En effet ou bien celui \347i est de degr\351 strictement sup\351rieur \340 trois auque l cas\nle premier terme est n\351gligeable devant le second et le seco nd ne tend pas vers 0. Ou bien le le polynome est de degr\351 inf\351r ieur ou \351gal \340 deux et donc le second terme devient n\351gligeab le par rapport au premier. Ce premier terme n'\351tant lui non plus pa s sommable car ne tendant pas vers 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Pn := a*n^3+b*n^2+c*n+d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PnG,** &%\"aG\"\"\")%\"nG\"\"$F(F(*&%\"bGF()F*\"\"#F(F(*&%\"cGF(F*F(F(%\"dGF( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "un := (n^7+3*n^6)^(1/7) -Pn^(1/3);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#unG,&* $),&*$)%\"nG\"\"(\"\"\"F-*&\"\"$F-)F+\"\"'F-F-#F-F,F-F-*$),**&%\"aGF-) F+F/F-F-*&%\"bGF-)F+\"\"#F-F-*&%\"cGF-F+F-F-%\"dGF-#F-F/F-!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "On va r\351aliser un d\351veloppe ment asymptotique \340 l'ordre 2 de la quantit\351 u_n car on sait qu' \340 partir de l'ordre 1/n^2 tous les termes sont sommables." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dev := series(un, n = infinity, 2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$devG,,*&,&\"\"\"F(*$)%\"aG#F(\"\"$F (!\"\"F(%\"nGF(F(#F-\"\"(F(*(F-F.F+#!\"#F-%\"bGF(F.*&,&#\"#F\"#\\F.*&F *F(,&*(F-F.%\"cGF(F+F.F(*(\"\"*F.F5\"\"#F+F4F.F(F.F(F/F.F(-%\"OG6#*&F( F(*$)F/FAF(F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "devTrunc := convert(dev, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)devTr uncG,**&,&\"\"\"F(*$)%\"aG#F(\"\"$F(!\"\"F(%\"nGF(F(#F-\"\"(F(*(F-F.F+ #!\"#F-%\"bGF(F.*&,&#\"#F\"#\\F.*&F*F(,&*(F-F.%\"cGF(F+F.F(*(\"\"*F.F5 \"\"#F+F4F.F(F.F(F/F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " sol := solve(\{coeffs(devTrunc, n)\}, \{a, b, c, d\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<&/%\"aG\"\"\"/%\"bG#\"\"*\"\"(/%\"cG#!#a\" #\\/%\"dGF4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Nous allons v\351r ifier nos solutions sur (au moins) un example." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "u2n := subs(sol, un);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$u2nG,&*$),&*$)%\"nG\"\"(\"\"\"F-*&\"\"$F-)F+\"\"'F-F -#F-F,F-F-*$),**$)F+F/F-F-*&#\"\"*F,F-*$)F+\"\"#F-F-F-*&#\"#a\"#\\F-F+ F-!\"\"%\"dGF-#F-F/F-FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d := 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sum2 := sum(u2n, n = 0 .. infinity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sum2G-%$sumG6$,&*$),&*$)%\"nG \"\"(\"\"\"F0*&\"\"$F0)F.\"\"'F0F0#F0F/F0F0*$),**$)F.F2F0F0*&#\"\"*F/F 0*$)F.\"\"#F0F0F0*&#\"#a\"#\\F0F.F0!\"\"F@F0#F0F2F0FE/F.;\"\"!%)infini tyG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(sum2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+La^f;!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}}{MARK "13 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }