{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 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 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 "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 "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 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 1 0 0 0 0 0 0 }0 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 1 0 0 0 0 0 0 }0 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 1 0 0 0 0 0 0 }0 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 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 16 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Jan K\344stle \+ \+ 24.11.97" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 6 "Folgen" }}}{SECT 1 {PARA 263 "" 0 "" {TEXT 258 29 "Verschiedene Arten von Folgen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 15 "monotone Folge:" }}{PARA 0 "" 0 "" {TEXT -1 90 "E ine Folge ist monoton, wenn die \304nderrungsrate entweder nur positiv oder nur negativ ist." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f [1]:=i->i^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\":6#%\"i G6\"6$%)operatorG%&arrowGF+*$9$\"\"#F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{seq([i,f[1](i)],i=1..10)\},style=point);" }}} {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "oszillierende Folge (alterniere nde Folge)" }}{PARA 0 "" 0 "" {TEXT -1 30 "Die Folge pendelt hin und h er." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f[2]:=i->sin(i);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"#%$sinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{seq([i,f[2](i)],i=1..20)\},style=p oint);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 17 "beschr\344nkte Folge " }}{PARA 0 "" 0 "" {TEXT -1 70 "Eine beschr\344nkte Folge geht \374be r eine bestimmte Schranke nicht hinaus." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f[3]:=i->4-2/i;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"fG6#\"\"$:6#%\"iG6\"6$%)operatorG%&arrowGF+,&\"\"%\"\"\"*$9$!\"\"! \"#F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{seq([i,f[ 3](i)],i=1..20)\},style=point);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 19 "unbeschr\344nkte Folge" }}{PARA 0 "" 0 "" {TEXT -1 79 "Die Folge s teigt oder sinkt, ohne eine Schranke nach oben oder unten zu haben. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f[4]:=i->i;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"%:6#%\"iG6\"6$%)operatorG%&arrowGF+ 9$F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{seq([i,f[4 ](i)],i=1..20)\},style=point);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 17 "konvergente Folge" }}{PARA 0 "" 0 "" {TEXT -1 55 "Eine konvergente Folge n\344hert sich einem Grenzwert an.." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "f[5]:=i->1-1/i;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"fG6#\"\"&:6#%\"iG6\"6$%)operatorG%&arrowGF+,&\"\"\"F0*$9$!\"\"F3F +F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(\{seq([i,f[5](i )],i=1..20)\},style=point);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 16 " divergente Folge" }}{PARA 0 "" 0 "" {TEXT -1 30 "Die Folge hat keinen \+ Grenzwert" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f[6]:=i->i;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"':6#%\"iG6\"6$%)operator G%&arrowGF+9$F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot( \{seq([i,f[6](i)],i=1..20)\},style=point);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Konvergente Folgen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Wie kann man zeigen, da\337 ab einem bestimmten " } {XPPEDIT 18 0 "n[0]" "&%\"nG6#\"\"!" }{TEXT -1 6 " alle " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 37 " in einem bestimmten Bereich liegen \+ ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=n->1/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"nG6\"6$%)operatorG%&arrowGF(*$9$ !\"\"F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Der Grenzwert f\374r die Folge " }{XPPEDIT 18 0 "f(n)=1/n" "/-%\"fG6#%\"nG*&\"\"\"\"\"\"F& !\"\"" }{TEXT -1 5 " ist:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit(f(n),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Ab welchem " }{XPPEDIT 18 0 "n[ 0]" "&%\"nG6#\"\"!" }{TEXT -1 69 " liegen alle n\264s in einem bestimm ten, beliebig kleine Bereich (Hier: " }{XPPEDIT 18 0 "1/1000" "*&\"\" \"\"\"\"\"%+5!\"\"" }{TEXT -1 5 " ) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "epsilon:=1/1000;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %(epsilonG#\"\"\"\"%+5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s olve(abs(f(n)) " 0 "" {MPLTEXT 1 0 14 "with(student):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 61 "pic1:=plot(\{epsilon,-epsilon,f(n)\},n=100.. 10000,color=black):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "l1:=leftbox (1/1000,x=0..10000,1,color=green,shading=green):l2:=leftbox(-1/1000,x= 0..10000,1,color=green,shading=green):\ntext:=textplot([500,0.0007, \264epsilon\264]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(l1,l2 ,pic1,text);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 160 }{VIEWOPTS 1 1 0 1 1 1803 }