{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 16 0 0 0 0 0 0 1 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 -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 }1 0 0 0 6 6 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 }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 }1 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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 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 {SECT 1 {PARA 3 "" 0 "" {TEXT 257 7 "Quelle:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 274 "Dateiname: s2231097.mws\nDateigr\366\337e: 14 \+ KB\nName: Mike & Martin\nSchule: Isolde-Kurz-Gymnasium\nKlasse: 12\nDa tum: 24.10.97\nFach: Mathematik\nThema: 2. Konvrgenz-Kriterium\nStichw \366rter: Beweis von Cauchy an einer Beispielfolge\nKurzbeschreibung: \+ Beweis durch vollst\344ndige Induktion\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 259 "" 0 "" {TEXT -1 8 "23.10.97" }}{PARA 0 "" 0 "" {TEXT -1 5 "M & M" }}{PARA 259 "" 0 "" {TEXT 256 42 "Das zweite Konver genz-Kriterium ( Cauchy )" }}{PARA 260 "" 0 "" {TEXT -1 41 "\"\"Mitsch rieb\" von Mathestunde am 20.10.97" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x[n]:=1/2*(x[n-1]+x[n-2 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#%\"nG,&&F%6#,&F'\"\"\" !\"\"F,#F,\"\"#&F%6#,&F'F,!\"#F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "x[0]:=0;x[1]:=1;\np[0]:=[0,0];p[1]:=[1,1];\nfor n fr om 2 to 10 \n do x[n]:=1/2*(x[n-1]+x[n-2]):\n p[n]:=[n,x[n]]:\nod ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"!7$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\" \"\"7$F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"##\"\"\"F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"#7$F'#\"\"\"F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$#F'\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"$7$F'#F'\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%#\"\"&\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"%7$F'#\"\"&\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"xG6#\"\"&#\"#6\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6# \"\"&7$F'#\"#6\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'# \"#@\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"'7$F'#\"#@\"# K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"(#\"#V\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"(7$F'#\"#V\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\")#\"#&)\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\")7$F'#\"#&)\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"*#\"$r\"\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"\"*7$F'#\"$r\"\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"#5#\"$T$\"$7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#\"#57$F'#\"$T$\"$7&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot(\{seq(p[i],i=0..10)\},x,style=point);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 303 "Die Folge ist ja wie ersichtlich nicht monoton, sondern \+ oszilliert. Aber ist sie trotzdem konvergent? Wenn ja, dann aber nicht nach dem 1. Kgz.-Kriterium, denn das schreibt ja neben der Beschr\344 nktheit auch die Monotonie vor. Wir brauchen also ein anderes Kgz.-Kri t. Das 2. Kgz.-Krit. lautet nach Cauchy:" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "x[n]" "&%\" xG6#%\"nG" }{TEXT -1 13 " ) kgt. <=> " }{XPPEDIT 18 0 "V[n[0]] " "&% \"VG6#&%\"nG6#\"\"!" }{TEXT -1 3 " " }{XPPEDIT 18 0 "A[(n,m)>=n[0] \+ " "&%\"AG6#1&%\"nG6#\"\"!6$F'%\"mG" }{TEXT -1 3 " " }{XPPEDIT 18 0 " A[epsilon>0]" "&%\"AG6#2\"\"!%(epsilonG" }{TEXT -1 7 " : " } {XPPEDIT 18 0 "abs(x[n]-x[m]) " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Bei der Betrachtung der Werte kommen wir zu folgender Vermutung :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(x[n+1]-x[n])=1/2^n" "/-%$absG6#,&&%\"xG6#,&%\"nG\" \"\"\"\"\"F,F,&F(6#F+!\"\"*&\"\"\"F,)\"\"#F+F0" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Dies vers uchen wir nun durch die vollst\344ndige Induktion zu beweisen. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "Induktionsannahme:=abs(x[n+1]-x[n])=1/2^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2InduktionsannahmeG/-%$absG6#,&&% \"xG6#,&%\"nG\"\"\"F/F/F/&F+6#F.!\"\"*$)\"\"#F.F2" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 17 "Induktionsanfang:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "x[0]:=0;\nx[1]:=1;\nfor n from 2 to 6 \n do x[n]:=1/ 2*(x[n-1]+x[n-2]):\nod;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6# \"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"##\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$#F'\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%#\"\"&\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"x G6#\"\"&#\"#6\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"'#\" #@\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for n from 0 to 5 \n do Induktionsannahme:\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\" \"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\"%F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"\")F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"#;F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"\"\"#KF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Unsere Induktionsannahme scheint wohl zu stimme n. Denn das sieht ja ganz vern\374nftigt aus." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Induktionsannahme;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6# ,&&%\"xG6#,&%\"nG\"\"\"F-F-F-&F)6#F,!\"\"*$)\"\"#F,F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(n=n+1,Induktionsannahme);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG6#,&&%\"xG6#,&%\"nG\"\"\"\"\"# F-F-&F)6#,&F,F-F-F-!\"\"*$)F.F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x[n]:=1/2*(x[n-1]+x[n-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#%\"nG,&&F%6#,&F'\"\"\"!\"\"F,#F,\"\"#&F%6#,&F' F,!\"#F,F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x[n+2]:=subs( n=n+2,x[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#,&%\"nG\"\"\" \"\"#F),&&F%6#,&F(F)F)F)#F)F*&F%6#F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x[n]:='x[n]':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Induktionsannahme;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$absG 6#,&&%\"xG6#,&%\"nG\"\"\"F-F-F-&F)6#F,!\"\"*$)\"\"#F,F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Wir erhalten die Induktionsannahme als Er gebnis und haben diese damit bewiesen. " }}{PARA 0 "" 0 "" {TEXT -1 53 "Damit ist au\337erdem gezeigt, da\337 folgendes stimmt: " } {XPPEDIT 18 0 "abs(x[n+1]-x[n]) = 1/(2^n)" "/-%$absG6#,&&%\"xG6#,&%\"n G\"\"\"\"\"\"F,F,&F(6#F+!\"\"*&\"\"\"F,)\"\"#F+F0" }{TEXT -1 4 " < " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "Aber es gilt noch endg\374ltig zu beweisen, ob die Folge auch \+ wirklich konvergent ist. D.h. im Klartext, da\337 wir nun \374berpr \374fen, ob das 1. Konvergenz-Kriterium zutrift, oder nicht. Also ob . .. wahr ist:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 "(" }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT -1 13 " ) \+ kgt. <=> " }{XPPEDIT 18 0 "V[n[0]] " "&%\"VG6#&%\"nG6#\"\"!" }{TEXT -1 3 " " }{XPPEDIT 18 0 "A[n>=n[0] " "&%\"AG6#1&%\"nG6#\"\"!F'" } {TEXT -1 3 " " }{XPPEDIT 18 0 "A[epsilon>0]" "&%\"AG6#2\"\"!%(epsilo nG" }{TEXT -1 7 " : " }{XPPEDIT 18 0 "abs(x[n]-g]) " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "x[n]:=rsolve(\{x(n)=1/2*(x(n-1)+x(n-2)),x(0)=0,x(1) =1\},x(n)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#%\"nG,&)#!\" \"\"\"#F'#!\"#\"\"$#F,F/\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Hier haben wir eine allgemeine Schreibweise von " }{XPPEDIT 18 0 "x[n ]" "&%\"xG6#%\"nG" }{TEXT -1 58 " . Daraus wird auch der Grenzwert der Funktion deutlich ( " }{XPPEDIT 18 0 "g=2/3)" "/%\"gG*&\"\"#\"\"\"\" \"$!\"\"" }{TEXT -1 4 " ). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g:=2/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG#\"\"#\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x[n+1]:=subs(n=n+1,x[n]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#,&%\"nG\"\"\"F)F),&)#!\"\"\" \"#F'#!\"#\"\"$#F.F1F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "z u_beweisen:=x[n+1]-x[n]=(-1)^n/2^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%,zu_beweisenG/,&)#!\"\"\"\"#,&%\"nG\"\"\"F-F-#!\"#\"\"$)F(F,#F*F0*& )F)F,F-)F*F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand(zu _beweisen);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)#!\"\"\"\"#%\"nG*&)F& F(\"\"\")F'F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 518 "Damit haben w ir ein feines Ergebnis, das uns die Gewi\337heit gibt, da\337 die Funk tion konvergent ist und das Konvergenzkriterium von Cauchy wohl zu sti mmen scheint; wir konnten auf jedenfall nicht das Gegenteil beweisen. \+ Interessant, da\337 Maple noch gar nicht daran denken w\374rde, den Be weis richtig zu nennen. Hier sieht man wieder wie \"doof\" Maple manch mal sein kann. Denn es beherrscht noch nicht einmal die Potenzrechenre geln der Klasse 8 (oder so \344hnlich; lang-lang isch her), aber das s ind ja fast schon \374bliche Sp\344\337e." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "evalb((-1/2)^n = (-1)^n/(2^n));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Daf\374r kann Maple \374ber \+ 'rsolve' den genauen Folgenwert einer Zahl benennen, auch nicht schlec ht!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "xn:=rsolve(\{x(n)=1/2*(x(n-1)+x(n-2 )),x(0)=0,x(1)=1\},x(n),makeproc);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#xnG:6#%\"nG6%%\"fG%\"iG%\"jG6\"F,@'29$\"\"!C%?(8%F0\"\"\"F4%%trueG> &8$6#F3-%#opG6$,&\"\"#F4F3!\"\"7$F0F4?(F3F?F?F/F5C$>&F86#F>-%%evalG6#- %%subsG6%/F/F3/%*_rsolve/aGF8,&&FN6#F4F?&FN6#F0F>?(8&F0F4F4F5>&F86#FU& F86#,&FUF4F4F4FD2F/F>-F;6$,&F/F4F4F4F@C%?(F3F0F4F4F5>F7-F;6$,&F3F4F4F4 F@?(F3F0F4,&F/F4!\"#F4F5C$>FD-FG6#-FJ6%FLFM,&FR#F4F>FPFjo?(FUF0F4F4F5> FWFYFDF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "for i from 0 \+ to 10 \n do xn(i):\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#6\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#@\"#K " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#V\"#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#&)\"$G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$r\" \"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$T$\"$7&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Das sieht ja vorne aus - klar - mu\337 es ja a uch. Wenn's interessiert, das ist der Wert der Folge bei " }{XPPEDIT 18 0 "n=753" "/%\"nG\"$`(" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "xn(753);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6##\"^yJ$) y07$[t9\"=?_]=#y_p^\"fCc1.VQ(Gw`i-&[\\E+OvL+&*y\"f`:$\\l$Qv%y9J8%>7!y1 L+i4ijM/[vA$R3&R7YCg,IO>#HXd`ifQ#GQ@5i_cs<+=)42`h(3C2.$z:\"^y'\\#o3oC- @\"H=q,'\\+V9V&>l? KT)*ei#f=pOS-X/HQzh.Q%*yNU2K:$*y%)eE+FZ1'HUJh3Y&*oB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Tolle Sache, was! Hier d\374rften wir also den Grenzwert schon ganz sch\366n nah gekommen sein. 1-2-3-Test! " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(\",250);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#$\"ez*)f\\4f5)e3dP2ommmmmmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmm'!$]#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "OK. Maple \+ hat sich wieder rehabiliert. So dat sollte f\374r heute jen\374gen. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }