{VERSION 3 0 "IBM INTEL NT" "3.0" } {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 "Helvetica" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Helvetica" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 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 "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 "Maple Plot" 0 13 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 "" 0 256 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 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 22 "Folgen und Grenzwerte\n" } {TEXT 257 41 " Oktober 98\nFabian Hust, VHumorus@aol.com" }}{PARA 0 " " 0 "" {TEXT 258 25 "Die verschiedenen Folgen:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 86 "plot(1/n,n=0..10,style=point,symbol=circle,title=`k onvergente Folge (mit Grenzwert)`);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7gn7$$\"1+++;arz@!#;$\"19I8j[v(e%!# :7$$\"1+++P.D)H#F*$\"1-/X'4O6N%F-7$$\"1+++e_y;CF*$\"1^Nv')zsPTF-7$$\"1 +++z,KNDF*$\"1yWQ+]FWRF-7$$\"1+++*4bQl#F*$\"1:#=6C.\"oPF-7$$\"1+++T\\# 4*GF*$\"1d3_J05fMF-7$$\"1+++\"y%*z7$F*$\"1Y-J!GOp>$F-7$$\"1+++kW8-OF*$ \"13:t!zJhx#F-7$$\"1+++XTFwSF*$\"1pQns1A`CF-7$$\"1+++3Q\\4YF*$\"1W6e2c Vp@F-7$$\"1+++pMrU^F*$\"1$H'4n()\\W>F-7$$\"1,++JJ$fn&F*$\"1@wXV[#=w\"F 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C$\"1M)3=1kV:\"!#77$$\"1++]i_QQgFC$\"114*o[%[H8Fet7$$\"1,+D\"y%3TiFC$ \"1P(>'[7><:Fet7$$\"1++]P![hY'FC$\"1G#y:hl\"[dQpLx>Fet7$$\"1+++v.I%)oFC$\"1_EzHD:YAFet7$$\"1mm\"zpe*zqFC$\"1Ih(eM (f7DFet7$$\"1,++D\\'QH(FC$\"1*y8T`*GIGFet7$$\"1LLe9S8&\\(FC$\"14dTP\"f e:$Fet7$$\"1,+D1#=bq(FC$\"1#e)o5>RDNFet7$$\"1LLL3s?6zFC$\"1t(*RSv: " 0 "" {MPLTEXT 1 0 22 "limit(n^4,n=infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Auch das hat Maple richtig erkannt: Strebt " }{XPPEDIT 18 0 "n^ 4" "6#*$%\"nG\"\"%" }{TEXT -1 57 " gegen unendlich, so gibt es keinen \+ Grenzwert (unendlich " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT -1 2 ")." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(n,n,s tyle=point,symbol=circle,title=`streng monotone Folge (ohne Grenzwert) `);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVES G6$7S7$$!#5\"\"!F(7$$!1nmm;p0k&*!#:F,7$$!1LL$3s%HaF.FK7$$!1******\\$*4)*\\F.FN7$$!1+++]_&\\c%F.FQ7$$!1+++]1aZ 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"" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 0 "" 0 "" {TEXT -1 143 "Den Grenzwert kann man sich von Maple ausrechnen lassen, man k ann ihn aber auch \"erahnen\". Hierzu ist es sinnvoll, eine Zahlenfolg e auszugeben." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "seq(1/n,n=1 ..30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6@\"\"\"#F#\"\"##F#\"\"$#F#\" \"%#F#\"\"&#F#\"\"'#F#\"\"(#F#\"\")#F#\"\"*#F#\"#5#F#\"#6#F#\"#7#F#\"# 8#F#\"#9#F#\"#:#F#\"#;#F#\"#<#F#\"#=#F#\"#>#F#\"#?#F#\"#@#F#\"#A#F#\"# B#F#\"#C#F#\"#D#F#\"#E#F#\"#F#F#\"#G#F#\"#H#F#\"#I" }}}{PARA 0 "" 0 " " {TEXT -1 203 "Bei diesem Beispiel kann man ohne Probleme erkennen, d a\337 der Grenzwert wohl 0 ist, schlie\337lich wird ja der Bruch immer kleiner. Wir k\366nnen Maple das (wohlgemerkt einfachste) Beispiel au ch rechnen lassen." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Grenzw ert:=limit(1/n,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Gren zwertG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Der Befehl " }{TEXT 259 5 "limit" }{TEXT -1 174 " kann also die Funktion auf ihr Verhalten in \+ der Unendlichen untersuchen. Das 'n' mu\337 hierf\374r nat\374rlich al s unendlich definiert werden. Maple benutzt den englischen Ausdruck " }{TEXT 260 7 "infnity" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "if Grenzwert=infinity then `divergente Folge` fi;\ni f Grenzwert=0 then `Nullfolge (konvergente Folge)`\nelse `Die Folge ha t den Grenzwert:`:Grenzwert fi;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% >Nullfolge~(konvergente~Folge)G" }}}{PARA 0 "" 0 "" {TEXT -1 190 "Mit \+ einer einfachen 'If-Schleife' l\344\337t sich \374berpr\374fen, ob ein e Folge einen Grenzwert g hat, eine Nullfolge ist (der Grenzwert ist 0 ) oder gar unendlich ist. Ein paar Beispiele:\nBeispiel 1: " } {XPPEDIT 18 0 "(n+1)/(2*n)" "6#*&,&%\"nG\"\"\"\"\"\"F&F&*&\"\"#F&F%F&! \"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Grenzwert:=limit((n +1)/(2*n),n=infinity):\nif Grenzwert=infinity then `divergente Folge` \+ fi;\nif Grenzwert=0 then `Nullfolge (konvergente Folge)`\nelse `Die Fo lge hat den Grenzwert:`:Grenzwert fi;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Die~Folge~hat~den~Grenzwert:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Beispiel 2: " } {XPPEDIT 18 0 "(n+n^2)/(n+1)" "6#*&,&%\"nG\"\"\"*$F%\"\"#F&F&,&F%F&\" \"\"F&!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "Grenzwert:=l imit(((n+n^2)/(n+1)),n=infinity):\nif Grenzwert=infinity then `diverge nte Folge` fi;\nif Grenzwert=0 then `Nullfolge (konvergente Folge)`\ne lse `Die Folge hat den Grenzwert:`:Grenzwert fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1divergente~FolgeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %=Die~Folge~hat~den~Grenzwert:G" }}{PARA 11 "" 1 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