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"Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 8 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 219 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "Ti mes" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT 222 14 "0 Introduction" }} {PARA 3 "" 0 "" {TEXT 222 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "MethodeNaive:=proc(n,a)\n" }{MPLTEXT 1 0 32 "evalf(sum((-1)^k*a( k),k=0..n));\n" }{MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"aGF%F%F%F%-I&evalfG%*protectedG6#-I$sumG6$F)I(_sysl ibGF%6$*&)!\"\"I\"kGF%\"\"\"-9%6#F3F4/F3;\"\"!9$F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 59 "Calcul de Log(2) en utilisant la somme d es (-1)^n / (n+1).\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "MethodeNaiv e(100,k->1/(k+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S,SGHr#[\\;uWf# *[uMU50#4%pJ2)p!#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Metho deNaive(1000,k->1/(k+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S.6jNkGq ![F\\'*f+Bo'38#)e:VYOp!#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(log(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+1=ZJp!#5" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 222 15 "1 Calcul des Pn" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 66 "Pn(X) est une variante du polyn\364me de Chebyshev Tn(X) d\351finie par\n" }{TEXT 223 62 "Pn(sin\262t) = cos(2nt), si t est un r\351el. (Pn(X) = Tn(1 - \+ 2X)).\n" }{TEXT 223 11 "P0(X) = 1,\n" }{TEXT 223 16 "P1(X) = 1 - 2X,\n " }{TEXT 223 36 "Pn+2(X) = 2(1 - 2X)Pn+1(X) - Pn(X).\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Pn:=proc(n) option remember;\n" }{MPLTEXT 1 0 12 "if n=0 then\n" }{MPLTEXT 1 0 8 " 1;\n" }{MPLTEXT 1 0 5 "else\n " }{MPLTEXT 1 0 18 " if n=1 then \n" }{MPLTEXT 1 0 17 " 1 -2*X;\n" }{MPLTEXT 1 0 10 " else\n" }{MPLTEXT 1 0 45 " ex pand(2*(1-2*X)*Pn(n-1)-Pn(n-2));\n" }{MPLTEXT 1 0 9 " fi;\n" } {MPLTEXT 1 0 4 "fi;\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%6#I)rememberGF%F%@%/9$\"\"!\"\"\"@%/F*F,,&F ,F,I\"XGF%!\"#-I'expandG%*protectedG6#,&*&F/F,-I#PnGF%6#,&F*F,!\"\"F,F ,\"\"#-F96#,&F*F,F1F,F " 0 "" {MPLTEXT 1 0 6 "Pn(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Pn(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",& \"\"\"F#I\"XG6\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Pn(5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.\"\"\"F#I\"XG6\"!#]*$F$\"\"#\"$+ %*$F$\"\"$!%?6*$F$\"\"%\"%!G\"*$F$\"\"&!$7&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 7(I\"GG6\"I\"HGF$I\"LGF$I\"PGF$I\"TGF$I\"UGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 51 "Une variante pour d\351finir la suite de polyn\364mes Pn\n" }{TEXT 223 55 "en utilisant la suite Tn d\351j\340 d\351finie d ans Maple dans \n" }{TEXT 223 51 "package orthopoly et la relation Pn( X) = Tn(1 - 2X)" }{MPLTEXT 1 0 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Pn2:=proc(n)\n" }{MPLTEXT 1 0 20 "expand(T(n,1-2*X));\n" } {MPLTEXT 1 0 4 "end;" }}{PARA 0 "> " 0 "" {TEXT 223 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%-I'expandG%*protectedG6#-I\"TGF %6$9$,&\"\"\"F/I\"XGF%!\"#F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Pn2(5);" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.\"\"\"F#I\"XG6\"!#]*$F$\"\"#\"$+%*$F$\"\"$!%?6*$F$\"\"% \"%!G\"*$F$\"\"&!$7&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 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Qy`(*pu!#;$\"1j))*fQ\"Q\"***!#;7$$\"1LLe9S8&\\(!#;$\"1Yi5TGx****!#;7$$ \"1n;aQ?V@v!#;$\"1C/t`!yb***!#;7$$\",D1Ixa(!#6$\"1GK=Zl*z(**!#;7$$\"1L $ek3GSd(!#;$\"1l)**H\\1p%**!#;7$$\"1mmT5hK+w!#;$\"0/Bp8+A!**!#:7$$\"1L LLe@#Hl(!#;$\"1s3!ys!er(*!#;7$$\"-D1#=bq(!#7$\"0^')!# 8$!2u!fCpdD$)>!#<7$$\"1LLe9tOc()!#;$!2(G%Gh6$>zP!#<7$$\"1m;H#e0I&))!#; $!0OHV^]_O&!#:7$$\"*&Qk\\*)!\"*$!1i>WzIJHo!#;7$$\"1nmT5ASg!*!#;$!1$ohn XF1F)!#;7$$\"1LL$3dg6<*!#;$!1k$*!#;$!1$*oT4X#p***!#;7$$\"/v=n=_W$*!#9$!1&4LNdsR** *!#;7$$\"1mmmmxGp$*!#;$!1M;ThKia**!#;7$$\"1M$eRA5\\Z*!#;$!1a7HnU]L$*!# ;7$$\"-D\"oK0e*!#7$!1&\\\\\"p'32'y!#;7$$\".Dcwz5j*!#8$!1h3\\LSM0o!#;7$ $\"*&oi\"o*!\"*$!1W1v_<1'\\&!#;7$$\".vV$R*G4$y*))!#<7$$ \"/D1k2/P)*!#9$\"2%\\2FyZ@tM!#=7$$\"0v=nj+U')*!#:$\"2w]w'*)f4%o\"!#<7$ $\".v$40O\"*)*!#8$\"2O@!R\"poV7$!#<7$$\"0DJ?Q?&=**!#:$\"2k&Q(\\s_?n%!# <7$$\"/voa-oX**!#9$\"1d1o)=Z6L'!#;7$$\"0vVt7SG(**!#:$\"0(e!f?Qd5)!#:7$ $\"#5!\"\"$\"#5!\"\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"- %%VIEWG6$;$\"\"!!\"\"$\"#5!\"\"%(DEFAULTG-%+AXESLABELSG6'-I#miG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibG6\"65Q\"x6\"/%'familyGQ!6\"/%%size GQ#106\"/%%boldGQ&false6\"/%'italicGQ%true6\"/%*underlineGQ&false6\"/% *subscriptGQ&false6\"/%,superscriptGQ&false6\"/%+foregroundGQ([0,0,0]6 \"/%+backgroundGQ.[255,255,255]6\"/%'opaqueGQ&false6\"/%+executableGQ& false6\"/%)readonlyGQ&false6\"/%)composedGQ&false6\"/%*convertedGQ&fal se6\"/%+imselectedGQ&false6\"/%,placeholderGQ&false6\"/%6selection-pla ceholderGQ&false6\"/%,mathvariantGQ'italic6\"Q!6\"-%%FONTG6%%(DEFAULTG %(DEFAULTG\"#5%+HORIZONTALG%+HORIZONTALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 224 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots):\n" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 222 16 "2 Mise en oeuvre" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 69 "Calcul de dn via la relation 2*dn = (3+2*2^(1/2))^n+(3-2*2^(1/2))^n \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dn:=proc(n)\n" }{MPLTEXT 1 0 43 "simplify(( (3+sqrt(8))^n+(3-sqrt(8))^n)/2);\n" }{MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%-I)simplifyG6$%*protectedGI( _syslibGF%6#,&),&\"\"$\"\"\"-I%sqrtGF(6#\"\")F09$#F0\"\"#),&F/F0F1!\" \"F5F6F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 28 "Calcul direct de dn:=Pn(-1)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dn2:=proc(n)\n" } {MPLTEXT 1 0 9 "local f;\n" }{MPLTEXT 1 0 21 "f:=unapply(Pn(n),X);\n" }{MPLTEXT 1 0 7 "f(-1);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6#I\"fGF%F%F%C$>8$-I(unapplyG6$%*protectedGI( _syslibGF%6$-I#PnGF%6#9$I\"XGF%-F*6#!\"\"F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "dn2(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%jL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dn(5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"%jL" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 30 "Calcul d es coefficients de Pn\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pk:=proc (n,k)\n" }{MPLTEXT 1 0 18 "coeff(Pn(n),X,k);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"kGF%F%F%F%-I&coeffG%*p rotectedG6%-I#PnGF%6#9$I\"XGF%9%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "pk(5,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "!$7&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 223 70 "Construction conjointe de pk et d e ck via les relations de r\351currence\n" }{TEXT 223 1 "p" }{TEXT 210 1 "0" }{TEXT 223 7 " = 1, p" }{TEXT 211 4 "k+1 " }{TEXT 223 3 "= p " }{TEXT 212 1 "k" }{TEXT 223 33 " * (k+n)*(k-n) / ((k+1/2)*(k+1))\n" }{TEXT 223 1 "c" }{TEXT 213 1 "0" }{TEXT 223 13 " = dn - p0, c" }{TEXT 214 1 "k" }{TEXT 223 6 " = - p" }{TEXT 215 1 "k" }{TEXT 223 4 " - c" }{TEXT 216 4 "k-1\n" }{TEXT 223 32 "Le programme renvoit le couple [" }{TEXT 217 1 "p" }{TEXT 218 1 "k" }{TEXT 219 2 ",c" }{TEXT 220 1 "k" } {TEXT 223 2 "]\n" }{TEXT 223 25 "qui d\351pend de n et de k.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "pkck:=proc(n,k)\n" }{MPLTEXT 1 0 15 "local p,c,d,i;\n" }{MPLTEXT 1 0 10 "d:=dn(n);\n" }{MPLTEXT 1 0 6 " p:=1;\n" }{MPLTEXT 1 0 8 "c:=d-p;\n" }{MPLTEXT 1 0 24 "for i from 0 to k-1 do \n" }{MPLTEXT 1 0 38 " p:=p*(i+n)*(i-n)/((i+1/2)*(i+1));\n" }{MPLTEXT 1 0 13 " c:=-p-c;\n" }{MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 7 "[p,c];\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"kGF%6&I\"pGF%I\"cGF%I\"dGF%I\"iGF%F%F%C'>8&-I#dnGF%6# 9$>8$\"\"\">8%,&F.F5F4!\"\"?(8'\"\"!F5,&9%F5F9F5I%trueG%*protectedGC$> F4*,F4F5,&F;F5F2F5F5,&F;F5F2F9F5,&F;F5#F5\"\"#F5F9,&F;F5F5F5F9>F7,&F4F 9F7F97$F4F7F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "pkck(2 ,2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\")\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 52 "SerieAlt prend en argument la suite (Ak) et re nvoit\n" }{TEXT 223 49 "une valeur approch\351e de la somme des (-1)^k * Ak." }{TEXT 221 2 " \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "SerieA lt:=proc(n,a)\n" }{MPLTEXT 1 0 17 "local S,k,p,c,d;\n" }{MPLTEXT 1 0 10 "d:=dn(n);\n" }{MPLTEXT 1 0 6 "p:=1;\n" }{MPLTEXT 1 0 8 "c:=d-p;\n" }{MPLTEXT 1 0 11 "S:=c*a(0);\n" }{MPLTEXT 1 0 23 "for k from 1 to n-1 do\n" }{MPLTEXT 1 0 40 " p:=p*(k-1+n)*(k-1-n)/((k-1/2)*(k));\n" } {MPLTEXT 1 0 13 " c:=-p-c;\n" }{MPLTEXT 1 0 17 " S:=S+c*a(k);\n" }{MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 12 "evalf(S/d);\n" }{MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"aGF%6'I\"S GF%I\"kGF%I\"pGF%I\"cGF%I\"dGF%F%F%C(>8(-I#dnGF%6#9$>8&\"\"\">8',&F/F6 F5!\"\">8$*&F8F6-9%6#\"\"!F6?(8%F6F6,&F3F6F:F6I%trueG%*protectedGC%>F5 *,F5F6,(FCF6F:F6F3F6F6,(FCF6F:F6F3F:F6,&FCF6#F:\"\"#F6F:FCF:>F8,&F5F:F 8F:>F<,&F " 0 "" {MPLTEXT 1 0 11 "Digits:=50;" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 69 "M\351thode naive de calcule des sommes partielles des s\351rie s altern\351es.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "MethodeNaive:= proc(n,a)\n" }{MPLTEXT 1 0 32 "evalf(sum((-1)^k*a(k),k=0..n));\n" } {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"a GF%F%F%F%-I&evalfG%*protectedG6#-I$sumG6$F)I(_syslibGF%6$*&)!\"\"I\"kG F%\"\"\"-9%6#F3F4/F3;\"\"!9$F%F%F%" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 225 6 "Log(2)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 59 "Calcul de Log(2) en utilisant la somme des (-1)^n / (n+1).\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ln2:=proc(n)\n" }{MPLTEXT 1 0 24 "SerieAlt(n,k->1/(k+ 1));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\" nG6\"F%F%F%-I)SerieAltGF%6$9$f*6#I\"kGF%F%6$I)operatorGF%I&arrowGF%F%* $,&F)\"\"\"F2F2!\"\"F%F%F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ln2(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SEgV8+b2ol " 0 "" {MPLTEXT 1 0 14 "evalf(log (2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SEgV8+b2ol " 0 "" {MPLTEXT 1 0 28 "MethodeNaive(10,k->1/ (k+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S,W:,W:,W:,W:,W:,W:,W:,Wlt !#]" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 225 2 "Pi" }}{PARA 0 "" 0 "" {TEXT 223 87 "Calcul d' une valeur \+ approch\351e du nombre Pi en utilisant la somme des (-1)^n / (2n+1).\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "SuitePi:=proc(n)\n" } {MPLTEXT 1 0 28 "4*SerieAlt(n,k->1/(2*k+1));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%,$-I)SerieAltGF%6$9 $f*6#I\"kGF%F%6$I)operatorGF%I&arrowGF%F%*$,&F*\"\"#\"\"\"F4!\"\"F%F%F %\"\"%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "SuitePi(25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$ \"S$4jP2cX7m%)p:CFDYQKz*e`EfTJ!#\\" }}}{PARA 11 "" 0 "" {TEXT 226 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S^P*Rpr>%)G]zKQVEYQKz*e`EfTJ!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "4*MethodeNaive(25,k->1/(2*k+1));" } }{PARA 11 "" 1 "" {XPPMATH 20 "$\"S6YZHU7*eq(zFw)o\"eug6g)GJXJ5$!#\\" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 225 17 "Constante d'Euler" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 32 "Calc ul de la constante d'Euler.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Ga mma:=proc(n)\n" }{MPLTEXT 1 0 9 "local l;\n" }{MPLTEXT 1 0 11 "l:=ln2( n);\n" }{MPLTEXT 1 0 38 "SerieAlt(n,k->-log(k+1)/(k+1))/l+l/2;\n" } {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6#I \"lGF%F%F%C$>8$-I$ln2GF%6#9$,&*&-I)SerieAltGF%6$F.f*6#I\"kGF%F%6$I)ope ratorGF%I&arrowGF%F%,$*&-I$logG6$%*protectedGI(_syslibGF%6#,&F.\"\"\"F CFCFCFB!\"\"FDF%F%F%FCF*FDFCF*#FC\"\"#F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Gamma(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SS ZGd)o(>:[Yg`[\\5;$Q@#)*\\m:sd!#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(gamma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S#*RfLf@/JC S#3!47lggG`,\\m:sd!#]" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 18 "4 la fon ction Zeta" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 70 "La fonction zeta de Riemann et d\351finie par f(s):=somm e n>=O 1/(n+1)^s\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "zeta:=proc(n, s)\n" }{MPLTEXT 1 0 40 "1/(1-2^(1-s))*SerieAlt(n,k->1/(k+1)^s);\n" } {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"s GF%F%F%F%*&,&\"\"\"F))\"\"#,&F)F)9%!\"\"F.F.-I)SerieAltGF%6$9$f*6#I\"k GF%F%6$I)operatorGF%I&arrowGF%F%*$),&F2F)F)F)T$F.F%F%6$F&F-F)F%F%F%" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=2;\n" }{MPLTEXT 1 0 11 " zeta(50,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SRwV_)*=#*=Dgkm^TskVE#[o1M\\k\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Zeta(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"So?,*\\*=#*=Dgkm^TskVE#[o1M\\k\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi^2/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"So?,*\\*=#*=Dgkm^TskVE#[o1M\\k\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=4;\n" }{MPLTEXT 1 0 11 "zeta(10,4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$ \"S51sK/Wy@S4E$)*Qz'o$RA*yMKK#3\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Zeta(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S (=>&4vuF!z;Tlp.g^\">Q6rLKK#3\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Pi^4/90);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S (=>&4vuF!z;Tlp.g^\">Q6rLKK#3\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=0;\n" }{MPLTEXT 1 0 11 "zeta(10,0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Zeta(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "$!SSgzu')\\6pI#HptYzxl4:_*y(* ****\\!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "$!S++++++++++++++++++++++++ ]!#]" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 26 "5 S\351ries \340 terme s positifs" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 225 48 "Methode de sommati on de s\351ries \340 termes positifs" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 66 "Methode naive pour \351valuer \+ la somme des s\351ries \340 termes positifs.\n" }{TEXT 223 1 " " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Sn:=proc(n,a)\n" }{MPLTEXT 1 0 26 " evalf(sum(1/k^2,k=1..n));\n" }{MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"aGF%F%F%F%-I&evalfG%*protectedG6#-I$sum G6$F)I(_syslibGF%6$*$I\"kGF%!\"#/F1;\"\"\"9$F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 1 "\n" }}{PARA 0 "" 0 "" {TEXT 223 121 "Methode pour \351valuer la somme des s\351ries \340 termes positifs utilisant la m \351thode naive pour \351valuer les s\351ries altern\351es.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "SnBis:=proc(n,a) option remember;\n " }{MPLTEXT 1 0 56 "evalf(sum((-1)^(m-1)*sum(2^k*a(2^k*m),k=0..n),m=1. .n));\n" }{MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$ I\"nG6\"I\"aGF%F%6#I)rememberGF%F%-I&evalfG%*protectedG6#-I$sumG6$F+I( _syslibGF%6$*&)!\"\",&I\"mGF%\"\"\"F4F7F7-F.6$*&)\"\"#I\"kGF%F7-9%6#*& F;F7F6F7F7/F=;\"\"!9$F7/F6;F7FEF%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bm:=proc(m,n,a)\n" }{MPLTEXT 1 0 33 "evalf(sum(2^k*a( 2^k*m),k=0..n));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"mG6\"I\"nGF%I\"aGF%F%F%F%-I&evalfG%*protectedG6#-I $sumG6$F*I(_syslibGF%6$*&)\"\"#I\"kGF%\"\"\"-9&6#*&F2F59$F5F5/F4;\"\"! 9%F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 124 "Methode pour \351va luer la somme des s\351ries \340 termes positifs utilisant la nouvelle m\351thode pour \351valuer les s\351ries altern\351es.\n" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "SeriePos:=proc(n,a)\n" }{MPLTEXT 1 0 28 "Ser ieAlt(n,m->bm(m+1,n,a));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I\"aGF%F%F%F%-I)SerieAltGF%6$9$f*6#I\"mGF%F%6 $I)operatorGF%I&arrowGF%F%-I#bmGF%6%,&F*\"\"\"F5F5T$T&F%F%6&F$F*F&9%F% F%F%" }}}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 225 14 "Somme des 1/k\262" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT 223 15 "Somme des 1/k\262\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=10;\n" }{MPLTEXT 1 0 13 "a:=k->1/k^2;\n" } {MPLTEXT 1 0 15 "SeriePos(n,a);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"kG6\"F%6$I)operatorGF%I&arro wGF%F%*$9$!\"#F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S**)zjXzuv(oQ !fUD(*f(=0Xpy38W;!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Sn (10,k->1/k^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S%4$)e&yy " 0 "" {MPLTEXT 1 0 20 "SnBis(1 0,k->1/k^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S)GCWw91C0or&*H>'=p ;][.gb7N;!#\\" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi^2/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"So?,*\\*=#*=Dgkm^TskVE#[o1M\\k\"!#\\" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 26 "Somme des (-1)^(n+1) / n\262\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "SerieAlt(25,k->1/(k+1)^ 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"Svukm+Q%4(*RBrg7-P#=K6CM.nC#)! #]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Pi^2/12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"SS.1&\\Z4Yf7IK$e2iB=K6CM.nC#)!#]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 223 20 "Somme des 1/(2n+1)\262\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a:=k->1/(2*k-1)^2;\n" }{MPLTEXT 1 0 7 "n:=10;\n" }{MPLTEXT 1 0 14 "SeriePos(n,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"kG6\"F%6$I)operatorGF%I&arrowGF%F%*$,&9$\"\"#!\"\" \"\"\"!\"#F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S.W<0CGk*zgw\"=16a< " 0 "" {MPLTEXT 1 0 14 "evalf(Pi^2/8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S^!fU7U\">*)=X)\\P6VNF)ph8]0qL7!#\\" }}}{PARA 0 "" 0 " " {XPPEDIT 20 0 "Typesetting:-mrow(Typesetting:-mn(\"1.233700550136169 8273543113749845188919142124259051\", mathvariant = \"normal\"));" "-I %mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#mnGF$6$QT1.233 7005501361698273543113749845188919142124259051F'/%,mathvariantGQ'norma lF'" }}}{PARA 11 "" 0 "" {TEXT 226 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 225 15 "Somme des 1/k^4" }}{PARA 4 "" 0 "" {TEXT 225 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=15;\n" }{MPLTEXT 1 0 13 "a :=k->1/k^4;\n" }{MPLTEXT 1 0 14 "SeriePos(n,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"kG6\"F%6$I)o peratorGF%I&arrowGF%F%*$9$!\"%F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " $\"SiVi!*yJNVjF$y)*Hj/CxO6PBBB3\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=4;\n" }{MPLTEXT 1 0 11 "zeta(15,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SXYkkDj&Hg *R\"yF()3Bp:P6PBBB3\"!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(Pi^4/90);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S(=>&4vuF!z;Tl p.g^\">Q6rLKK#3\"!#\\" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 31 "6 Ex emples avec Pn(X) = (1-X)^n" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 25 "On prend Pn(X)= (1-X)^n,\n" }{TEXT 223 38 "Pn(X)= sum( (-1)^k*C(n,k)*X^k ,k=0..n)\n" }{TEXT 223 37 "d'o\371 pour n fix\351, pk:= (-1)^k C(n,k) \n" }{TEXT 223 46 "avec C(n,k) le coefficent binomial k parmi n.\n" } {TEXT 223 18 "dn= Pn(-1) = 2^n.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "SerieAlt2:=proc(n,a)\n" }{MPLTEXT 1 0 17 "local S,k,p,c,d;\n" } {MPLTEXT 1 0 8 "d:=2^n;\n" }{MPLTEXT 1 0 6 "p:=1;\n" }{MPLTEXT 1 0 8 " c:=d-p;\n" }{MPLTEXT 1 0 11 "S:=c*a(0);\n" }{MPLTEXT 1 0 23 "for k fro m 0 to n-1 do\n" }{MPLTEXT 1 0 35 " p:=(-1)^(k+1)*binomial(n,k+1); \n" }{MPLTEXT 1 0 13 " c:=-p-c;\n" }{MPLTEXT 1 0 19 " S:=S+c*a(k +1);\n" }{MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 12 "evalf(S/d);\n" } {MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"nG6\"I \"aGF%6'I\"SGF%I\"kGF%I\"pGF%I\"cGF%I\"dGF%F%F%C(>8()\"\"#9$>8&\"\"\"> 8',&F/F5F4!\"\">8$*&F7F5-9%6#\"\"!F5?(8%F@F5,&F2F5F9F5I%trueG%*protect edGC%>F4*&)F9,&FBF5F5F5F5-I)binomialG6$FEI(_syslibGF%6$F2FJF5>F7,&F4F9 F7F9>F;,&F;F5*&F7F5-F>6#FJF5F5-I&evalfGFE6#*&F;F5F/F9F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ln2bis:=proc(n)\n" }{MPLTEXT 1 0 25 "SerieAlt2(n,k->1/(k+1));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%-I*SerieAlt2GF%6$9$f*6#I\"kGF% F%6$I)operatorGF%I&arrowGF%F%*$,&F)\"\"\"F2F2!\"\"F%F%F%F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ln2bis(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SF+*>3_*pK)o\"e97KsT4`%*f0=ZJp!#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(log(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SEgV8+b2ol " 0 "" {MPLTEXT 1 0 18 "SuitePi2:= proc(n)\n" }{MPLTEXT 1 0 29 "4*SerieAlt2(n,k->1/(2*k+1));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%,$-I*S erieAlt2GF%6$9$f*6#I\"kGF%F%6$I)operatorGF%I&arrowGF%F%*$,&F*\"\"#\"\" \"F4!\"\"F%F%F%\"\"%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "SuitePi2(25);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SCQ4G8**)4 " 0 "" {MPLTEXT 1 0 10 "eval f(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S^P*Rpr>%)G]zKQVEYQKz*e`EfT J!#\\" }}}{PARA 0 "" 0 "" {TEXT 223 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 222 42 "7 Exemples avec Pn(X)=X^q (1-X)^q, 3n = 3q" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 36 "On prend Pn(X)=X^q (1-X)^q, 3n = 3q\n" } {TEXT 223 18 "dn = (-1)^q*2^(2q)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "SerieAlt3:=proc(q,a)\n" }{MPLTEXT 1 0 19 "local S,k,p,c,d,n;\n" } {MPLTEXT 1 0 8 "n:=3*q;\n" }{MPLTEXT 1 0 19 "d:=(-1)^q*2^(2*q);\n" } {MPLTEXT 1 0 6 "p:=0;\n" }{MPLTEXT 1 0 6 "c:=d;\n" }{MPLTEXT 1 0 11 "S :=c*a(0);\n" }{MPLTEXT 1 0 23 "for k from 1 to n-1 do\n" }{MPLTEXT 1 0 17 " if k=q then \n" }{MPLTEXT 1 0 14 " p:=1;\n" }{MPLTEXT 1 0 9 " else\n" }{MPLTEXT 1 0 20 " if k>q then\n" }{MPLTEXT 1 0 33 " p:=p*(k-1-n)/(k-q);\n" }{MPLTEXT 1 0 12 " \+ fi;\n" }{MPLTEXT 1 0 8 " fi;\n" }{MPLTEXT 1 0 16 " c:=-p-c; \+ \n" }{MPLTEXT 1 0 17 " S:=S+c*a(k);\n" }{MPLTEXT 1 0 4 "od;\n" } {MPLTEXT 1 0 12 "evalf(S/d);\n" }{MPLTEXT 1 0 5 "end;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"qG6\"I\"aGF%6(I\"SGF%I\"kGF%I\"pGF%I\"cGF%I \"dGF%I\"nGF%F%F%C)>8),$9$\"\"$>8(*&)!\"\"F2\"\"\")\"\"#,$F2F;F9>8&\" \"!>8'F5>8$*&FAF9-9%6#F?F9?(8%F9F9,&F0F9F8F9I%trueG%*protectedGC%@%/FI F2>F>F9@$2F2FI>F>*(F>F9,(FIF9F8F9F0F8F9,&FIF9F2F8F8>FA,&F>F8FAF8>FC,&F CF9*&FAF9-FF6#FIF9F9-I&evalfGFL6#*&FCF9F5F8F%F%F%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "ln2ter:=proc(n)\n" }{MPLTEXT 1 0 25 "SerieAl t3(n,k->1/(k+1));\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%-I*SerieAlt3GF%6$9$f*6#I\"kGF%F%6$I)ope ratorGF%I&arrowGF%F%*$,&F)\"\"\"F2F2!\"\"F%F%F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ln2ter(30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S&\\vB,]v!ol " 0 "" {MPLTEXT 1 0 14 "evalf(log(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SEgV8+b2ol " 0 "" {MPLTEXT 1 0 18 "SuitePi3:=proc(n)\n " }{MPLTEXT 1 0 29 "4*SerieAlt3(n,k->1/(2*k+1));\n" }{MPLTEXT 1 0 4 "e nd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"F%F%F%,$-I*SerieAlt3G F%6$9$f*6#I\"kGF%F%6$I)operatorGF%I&arrowGF%F%*$,&F*\"\"#\"\"\"F4!\"\" F%F%F%\"\"%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "SuiteP i3(10);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S?Newbl%>%Q_pITH%yY0z*e` EfTJ!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"S^P*Rpr>%)G]zKQVEYQKz*e`EfTJ!#\\" }}} {PARA 0 "" 0 "" {TEXT 223 0 "" }}{PARA 0 "" 0 "" {TEXT 223 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 222 26 "8 Comparaison des m\351thodes" } }{PARA 0 "" 0 "" {TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 223 89 "La m\351thode de Simpson est une m\351thode rapide de calcul de valeu r approch\351e d'int\351grales.\n" }{TEXT 223 82 "On peut l'applique p our calculer l'int\351grale de 0 \340 1 de 1/ 1 + x\262 qui vaut Pi/4. \n" }{TEXT 223 75 "La m\351thode pr\351sent\351e dans le texte donne u ne meilleur approximation de Pi.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Simpson:=proc(n,f,a,b)\n" }{MPLTEXT 1 0 13 "local k,S,j;\n" } {MPLTEXT 1 0 12 "j:=(b-a)/n;\n" }{MPLTEXT 1 0 6 "S:=0;\n" }{MPLTEXT 1 0 23 "for k from 0 to n-1 do\n" }{MPLTEXT 1 0 49 "S:=S+(f(a+k*j)+f(a+( k+1)*j)+4*f(a+(k+1/2)*j))/6;\n" }{MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 12 "evalf(j*S);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"nG6\"I\"fGF%I\"aGF%I\"bGF%6%I\"kGF%I\"SGF%I\"jGF%F%F%C&>8&* &,&9'\"\"\"9&!\"\"F39$F5>8%\"\"!?(8$F9F3,&F6F3F5F3I%trueG%*protectedG> F8,*F8F3-9%6#,&F4F3*&F;F3F/F3F3#F3\"\"'-FB6#,&F4F3*&,&F;F3F3F3F3F/F3F3 FF-FB6#,&F4F3*&,&F;F3#F3\"\"#F3F3F/F3F3#FS\"\"$-I&evalfGF>6#*&F/F3F8F3 F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->1/(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%6$I)operatorGF%I&arrowG F%F%*$,&\"\"\"F+*$9$\"\"#F+!\"\"F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "4*Simpson(1000,f,0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"SWzb!>53;N'pfVL-i%QKz*e`EfTJ!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "SuitePi(50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S'o* 4:<(>%)G]zKQVEYQKz*e`EfTJ!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"S^P*Rpr>%)G]zKQ VEYQKz*e`EfTJ!#\\" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G " }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }