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}{PSTYLE "subp roblem" -1 292 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "diagram" -1 293 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "dblnorm.mws" -1 294 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 2 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Item" -1 295 1 {CSTYLE "" -1 -1 "Lucida Sans" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 6 0 3 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 296 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 297 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 297 "" 0 "" {TEXT -1 36 "Lesson 15: Convergence of Sequences" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Sequence Convergence" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 382 "We have used sequences lots of times before. The sequence of estimates to the solution of an equation generated by Ne wton's Method is one. The sequence of estimates to the integral of a \+ function over an interval obtained by subdividing the interval into mo re and more subintervals is another. These are examples of potential ly infinite sequences. These are sequences we hope " }{TEXT 258 11 " converge " }{TEXT -1 94 " to the answer we seek, whether it be the s olution of an equation or the value of an integral." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Formally, a " }{TEXT 258 19 "sequence of numbers" }{TEXT -1 190 " is defined as a functi on f whose domain is the positive integers. The terms of the sequenc e are the values of the function. So for example the 10th term of the sequence f is f(10). " }}{PARA 0 "" 0 "converges to a limit" {TEXT -1 13 "A sequence f " }{TEXT 258 20 "converges to a limit" }{TEXT -1 121 " L if each interval containing L contains all but finitely m any terms of the sequence. In this case, we would write " }{XPPEDIT 18 0 "limit(f(n),n=infinity)=L" "6#/-%&limitG6$-%\"fG6#%\"nG/F*%)infin ityG%\"LG" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Th e Maple word " }{TEXT 256 5 "limit" }{TEXT -1 90 " can be used to c alculate many limits of sequences in a painfree manner. For example, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "limit((1+1/n)^n,n=infin ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The next theorems summarize many of the p roperties of convergent sequences. " }}{PARA 287 "" 0 "" {TEXT -1 14 " Theorem: If " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 33 " is an increasing sequence (ie, " }{XPPEDIT 18 0 "a[i]<=a[i+1]" "6#1& %\"aG6#%\"iG&F%6#,&F'\"\"\"F+F+" }{TEXT -1 9 " for all " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 10 "), then " }{XPPEDIT 18 0 "a[n]" "6#&% \"aG6#%\"nG" }{TEXT -1 57 " converges if there is an upper bound on the terms of " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 287 "" 0 "" {TEXT -1 15 "The orem: If " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 14 " c onverges to " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT -1 14 " converges to " } {XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "a[n] *b[n] " "6#*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F'F(" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "LM" "6#%#LMG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a[n]+b[n]" "6#,&&%\"aG6#%\"nG\"\"\"&%\"bG6#F'F(" }{TEXT -1 14 " conve rges to " }{XPPEDIT 18 0 "L + M" "6#,&%\"LG\"\"\"%\"MGF%" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "a[n]-b[n]" "6#,&&%\"aG6#%\"nG\"\"\"&%\"bG6#F '!\"\"" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "L - M" "6#,&%\"L G\"\"\"%\"MG!\"\"" }{TEXT -1 14 " . Also, if " }{XPPEDIT 18 0 "M<>0 " "6#0%\"MG\"\"!" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "a[n]/b[n]" "6# *&&%\"aG6#%\"nG\"\"\"&%\"bG6#F'!\"\"" }{TEXT -1 14 " converges to " } {XPPEDIT 18 0 "L/M" "6#*&%\"LG\"\"\"%\"MG!\"\"" }{TEXT -1 3 ". " }} {PARA 287 "" 0 "" {TEXT -1 13 "Theorem: If " }{XPPEDIT 18 0 "a[n] " " 6#&%\"aG6#%\"nG" }{TEXT -1 41 " is a sequence of positive numbers and " }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT -1 46 " is a sequen ce which converges to 0, then if " }{XPPEDIT 18 0 "a[n] <= b[n]" "6#1 &%\"aG6#%\"nG&%\"bG6#F'" }{TEXT -1 10 " for all " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 9 " , then " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\" nG" }{TEXT -1 16 " converges to 0." }}{PARA 287 "" 0 "" {TEXT -1 14 "T heorem: If " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 18 " is continuo us at " }{XPPEDIT 18 0 "x=L" "6#/%\"xG%\"LG" }{TEXT -1 7 ", and " } {XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 29 " is a sequence co nverging to " }{XPPEDIT 18 0 "L" "6#%\"LG" }{TEXT -1 21 ", then the se quence " }{XPPEDIT 18 0 "f(a[n])" "6#-%\"fG6#&%\"aG6#%\"nG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "f(L)" "6#-%\"fG6#%\"LG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 30 " Periodic Points of functions." }}{EXCHG {PARA 0 "" 0 "sequence of iterates" {TEXT -1 127 " Let f be a function , and let a be a point in the domain of f. If each value of f is in \+ the domain of f, we can generate the " }{TEXT 258 20 "sequence of iter ates" }{TEXT -1 30 " of a under f as follows: " }{XPPEDIT 18 0 "a[ 1] = f(a)" "6#/&%\"aG6#\"\"\"-%\"fG6#F%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2] = f(a[1])" "6#/&%\"aG6#\"\"#-%\"fG6#&F%6#\"\"\"" }{TEXT -1 17 ", and in general " }{XPPEDIT 18 0 "a[n+1]=f(a[n])" "6#/&%\"aG6#,&% \"nG\"\"\"F)F)-%\"fG6#&F%6#F(" }{TEXT -1 69 " for each positive intege r n. If n is a positive integer such that " }{XPPEDIT 18 0 "a[n]=a" "6#/&%\"aG6#%\"nGF%" }{TEXT -1 7 ", but " }{XPPEDIT 18 0 "a[i] <>a" " 6#0&%\"aG6#%\"iGF%" }{TEXT -1 96 " for for all positive integers i \+ < n, then a is called a periodic point of period n for f. " }}{PARA 0 "" 0 "fixed points" {TEXT -1 30 "Period one points are called " } {TEXT 258 12 "fixed points" }{TEXT -1 90 ". You can locate the fixe d points of a function by looking to see where the graphs of " } {XPPEDIT 18 0 "y = f(x) " "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 "and \+ " }{XPPEDIT 18 0 "y = x" "6#/%\"yG%\"xG" }{TEXT -1 90 " cross. For e xample, the cosine function has one fixed point, as we can see by plot ting." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(\{cos(x),x\},x=-Pi..Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!1++JZEfTJ!#:F(7$$!1#*pr)3PY+$ F*F,7$$!13E[*ysa)GF*F/7$$!1Y&))>2g9v#F*F27$$!1,@@J!flh#F*F57$$!1#QM::* H#[#F*F87$$!1]AhcI#yN#F*F;7$$!1'=d-FN*GAF*F>7$$!1&*GBP$Rc4#F*FA7$$!1Dj &f)3xi>F*FD7$$!1<(4AN*4E=F*FG7$$!1QQ*3**=dq\"F*FJ7$$!1\">jg@*>q:F*FM7$ $!1CZ'eYk>F*Fjr7$$\"1Zuyfix%4 #F*F]s7$$\"14q))fu.GAF*F`s7$$\"1@w'**=&>gBF*Fcs7$$\"1?3a=Wj\"[#F*Ffs7$ $\"1?c*)yu\"3i#F*Fis7$$\"1i-HnWIXFF*F\\t7$$\"1>RWhN.yGF*F_t7$$\"1t(*RS A20IF*Fbt7$$\"1++X]EfTJF*Fet-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F^u-F$6$7 en7$F($F]uF^u7$$!1'\\8!o[6tIF*$!1dR#\\7jl(**Fhn7$F,$!1h@plBO1**Fhn7$$! 1+)*4R\\0XHF*$!1@$G=_&[2)*Fhn7$F/$!1/?(R3-Qn*Fhn7$F2$!1tjRdke[#*Fhn7$F 5$!1#))3;*32`')Fhn7$F8$!1t%*)4+_U!zFhn7$F;$!1dVKLTd#3(Fhn7$F>$!1!=0%3d X;hFhn7$FA$!1BRr;:x5]Fhn7$FD$!1B/R/'R,#QFhn7$FG$!15L?OiQDDFhn7$FJ$!1Xj d$yO^M\"Fhn7$FM$\"1_hBip5rf!#>7$FP$\"1!*e)QR#[i8Fhn7$FS$\"1\"4??=#=YEF hn7$FV$\"1%ft'f*3Jx$Fhn7$FY$\"1vuI)H:B/&Fhn7$Ffn$\"1-2e(\\*[RgFhn7$Fjn $\"1V=%RO;$*3(Fhn7$F]o$\"1P!R#>U?/zFhn7$F`o$\"1f2))\\nIf')Fhn7$Fco$\"1 ^32!GO4B*Fhn7$Ffo$\"1UpZ8Nij'*Fhn7$$!1\"4gMblG)>Fhn$\"16p#*ya0/)*Fhn7$ Fio$\"1`tH@+.2**Fhn7$$!1/@Z/\"\\$yp!#<$\"1PID,7mv**Fhn7$F\\p$\"1%Q>$4> &*****Fhn7$$\"11t!3)GF;mF\\[l$\"1i.#)\\/7y**Fhn7$F`p$\"1o^XFhn$\"1q7lgk24)*Fhn7$Fcp$\"1,KPla1u'*Fhn7$Ffp$\"1;a.6AEj #*Fhn7$Fip$\"1X2bIh=u')Fhn7$F\\q$\"1LTpqx1YzFhn7$F_q$\"1bB(y$fc5rFhn7$ Fbq$\"1dw$GV'e[gFhn7$Feq$\"1/%3IU)o!*\\Fhn7$Fhq$\"1u!\\))[\"ysPFhn7$F[ r$\"1Lul!fl'3EFhn7$F^r$\"1gG.(ym:H\"Fhn7$Far$\"1ggeC[QdIF^p7$Fdr$!1X(f wBBxG\"Fhn7$Fgr$!19c4G8/bDFhn7$Fjr$!1.=qm$)zNQFhn7$F]s$!1(H!ec0I.]Fhn7 $F`s$!1#4B*z.N4hFhn7$Fcs$!1Hdan.I*4(Fhn7$Ffs$!1yl.w$y,!zFhn7$Fis$!13*G 2]PVn)Fhn7$F\\t$!1:DUS4+D#*Fhn7$F_t$!1:[-g))oa'*Fhn7$$\"1Y=#4!HbTHF*$! 1\"=4Vz'e+)*Fhn7$Fbt$!1u'y3`bp!**Fhn7$$\"1')[UXCLtIF*$!1tebk " 0 "" {MPLTEXT 1 0 32 "fix := fsolve(cos(x)=x,x,0..Pi);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fixG$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "" 0 "attracting fixed point" {TEXT -1 5 "An " }{TEXT 258 22 "attracting fixed point" }{TEXT -1 70 " is a fixed point a with \+ the property that for points b close to a," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "limit(b[n],n=infinity) = a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&limitG6$&%\"bG6#%\"nG/F*%)infinityG%\"aG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "whe re " }{XPPEDIT 18 0 "b[1 ]= b" "6#/&%\"bG6#\"\"\"F%" }{TEXT -1 8 " , \+ and " }{XPPEDIT 18 0 "b[n ]= f(b[n-1])" "6#/&%\"bG6#%\"nG-%\"fG6#&F%6 #,&F'\"\"\"F.!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n=2,3" "6$/%\" nG\"\"#\"\"$" }{TEXT -1 4 " ..." }}{PARA 0 "" 0 "repelling fixed point " {TEXT -1 4 "A " }{TEXT 258 21 "repelling fixed point" }{TEXT -1 87 " is a fixed point a with the property that for points b close \+ (but not equal) to a," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "limit(b[n],n=infinity) <> a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#0-%&limitG6$&%\"bG6#%\"nG/F*%)infinityG%\"aG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "whe re " }{XPPEDIT 18 0 "b[1 ]= b" "6#/&%\"bG6#\"\"\"F%" }{TEXT -1 8 " , \+ and " }{XPPEDIT 18 0 "b[n ]= f(b[n-1]) " "6#/&%\"bG6#%\"nG-%\"fG6#&F% 6#,&F'\"\"\"F.!\"\"" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "n = 2, 3" "6 $/%\"nG\"\"#\"\"$" }{TEXT -1 8 ", ... ." }}{PARA 0 "" 0 "" {TEXT -1 1 "~" }}{PARA 0 "" 0 "" {TEXT -1 90 "Here is a simple procedure to in vestigate periodic points and fixed points of a function." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "iterate := \+ proc(f,n,x)\n local a,i,s;\n a := evalf(x); \n s := a; \n \+ for i to n do a := f(a); \n s := s,a od\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "For example, to investigate whether the \+ fixed point of the cosine function is attracting or not, we can iterat e the function at a point near the fixed point. Using the fixed point \+ of the cos function," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fix := fsolve(cos(x)=x,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$fixG$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "iterate(cos,10,fix-1),fix;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6.$!+o'[\"4E!#5$\"+$zV:m*F%$\"+4an%o&F%$\"+.&psU)F%$\"+>u HamF%$\"+j`^myF%$\"+v&*>iqF%$\"+:T?3wF%$\"+ycqUsF%$\"+^$H)*[(F%$\"+7w \"QK(F%$\"+K8&3R(F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "The fixed point seems to be an attracting one. On the other hand if we look at the fixed points of 2x(1-x)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := x-> 2*x*(1-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR 6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&9$\"\"\",&F/F/F.!\"\"F/\"\"#F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fix := fsolve(f(x)=x,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fixG6$\"\"!$\"+++++]!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "iterate(f,10,fix[1]-.2);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6-$!\"#!\"\"$!#[F$$!&3U\"!\"%$!*GX*yo!\" )$!+pm(R3\"!\"($!+s.prB!\"&$!+y/.D6\"\"!$!+crQJD\"\"*$!+(=%e\"G\"\"#G$ !+cg\"\\G$\"#l$!+)pM\"e@\"$S\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "iterate(f,10,fix[1]+.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-$ \"\"\"!\"\"$\"#=!\"#$\"%_H!\"%$\")#R6;%!\")$\"+7DEf[!#5$\"+#fQg*\\F1$ \"+io****\\F1$\"+++++]F1F6F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "iterate(f,10,fix[2]+.4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$ \"+++++!*!#5$\"+++++=F%$\"++++_HF%$\"++#R6;%F%$\"+7DEf[F%$\"+#fQg*\\F% $\"+io****\\F%$\"+++++]F%F2F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "It seems that 0 is an repelling fixed point and that .5 is an att racting fixed point. Let's define a visual word to go with iterate. We have added a domain and range to allow you to determine the viewing w indow." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "viterate := proc (f,n,start,domain,range)\n local a, i, s, gra, gpl, fpl, ipl;\n \+ a := evalf(start);\n gra := [a,f(a)]; \n for i to n do a := f( a); \n gra := gra,[a,a],[a,f(a)];\n od:\n gpl := plot([gra],c olor=red);\n fpl := plot(f,domain,color=black);\n ipl := plot(x- >x,domain,color=blue);\n print(plots[display]([gpl,fpl,ipl],view=[d omain,range]));\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "viterate(x->2*x*(1-x),10,.8,-1..1 ,-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$777$$\"1+++++++!)!#;$ \"1+++++++KF*7$F+F+7$F+$\"1++++++_VF*7$F/F/7$F/$\"1++++#>g\"\\F*7$F3F3 7$F3$\"1+++Y%*e)*\\F*7$F7F77$F7$\"1+++-'*****\\F*7$F;F;7$F;$\"1+++++++ ]F*7$F?F?FAFAFAFAFAFAFAFAFA-%'COLOURG6&%$RGBG$\"*++++\"!\")\"\"!FI-F$6 $7S7$$!\"\"FI$!\"%FI7$$!1nmm;p0k&*F*$!1C\"Qu2NAu$!#:7$$!1LL$3s%HaF*$!1m\\ $R!yZv;FW7$$!1+++]$*4)*\\F*$!14%\\A\")R#*\\\"FW7$$!1+++]_&\\c%F*$!1^+p ytwH8FW7$$!1+++]1aZTF*$!1[!o))**\\N<\"FW7$$!1nm;/#)[oPF*$!1+At2xsP5FW7 $$!1MLL$=exJ$F*$!1H$!1svuytIw> !#=7$$\"1uN\")4i'\\F*7$$\"1mmm;WTAeF*$\"1mxyxlFF*7$$\"1LLLLbd Q()F*$\"1yAJ'f5Y?#F*7$$\"1++]i`1h\"*F*$\"1[0!)4q5P:F*7$$\"1++]P?Wl&*F* $\"1*obW+yMJ)Fit7$$\"\"\"FIFI-FC6&FEFIFIFI-F$6$7S7$FNFN7$FSFS7$FYFY7$F hnFhn7$F]oF]o7$FboFbo7$FgoFgo7$F\\pF\\p7$FapFap7$FfpFfp7$F[qF[q7$F`qF` q7$FeqFeq7$FjqFjq7$F_rF_r7$FdrFdr7$FirFir7$F^sF^s7$FcsFcs7$FhsFhs7$F]t F]t7$FbtFbt7$FgtFgt7$F]uF]u7$FbuFbu7$FiuFiu7$F^vF^v7$FcvFcv7$FhvFhv7$F ]wF]w7$FbwFbw7$FgwFgw7$F\\xF\\x7$FaxFax7$FfxFfx7$F[yF[y7$F`yF`y7$FeyFe y7$FjyFjy7$F_zF_z7$FdzFdz7$FizFiz7$F^[lF^[l7$Fc[lFc[l7$Fh[lFh[l7$F]\\l F]\\l7$Fb\\lFb\\l7$Fg\\lFg\\l7$F\\]lF\\]l-FC6&FEFIFIFF-%+AXESLABELSG6$ %!GFi`l-%%VIEWG6$;FNF\\]lF]al" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "This gives a nice visual tool to investig ate fixed points and periodic points of functions." }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "Problems" }}{EXCHG {PARA 288 "" 0 "" {TEXT -1 164 " Exercise: Use iterate or viterate to check more starting poin ts close the the fixed point of cos. Do you remain convinced that it is a repelling fixed point? " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fx := fsolve(cos(x)=x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# fxG$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "viterat e(cos,10,.1,-1..2,-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$777$$\"1+++++++5!#;$\"1+++`;/]**F*7$F+F+7$F +$\"1+++eR*\\W&F*7$F/F/7$F/$\"1+++fq'Qb)F*7$F3F37$F3$\"1+++PmEflF*7$F7 F77$F7$\"1+++>5$[#zF*7$F;F;7$F;$\"1+++!o#z?qF*7$F?F?7$F?$\"1+++P.,NwF* 7$FCFC7$FC$\"1+++ij>CsF*7$FGFG7$FG$\"1+++*e!3-vF*7$FKFK7$FK$\"1+++?.Z: tF*7$FOFO7$FO$\"1+++C%=9W(F*-%'COLOURG6&%$RGBG$\"*++++\"!\")\"\"!Ffn-F $6$7S7$$!\"\"Ffn$\"1)R\"oeI-.aF*7$$!1*****\\P&3Y$*F*$\"1*4MB^L8%fF*7$$ !1++Dcx6x()F*$\"1$Gt4)48*Q'F*7$$!1,+]iTDP\")F*$\"1Pk4LQ&z'oF*7$$!1**** \\P\"\\J\\(F*$\"15ns%yc:K(F*7$$!1++DJa5_oF*$\"1Ofv\\#eGu(F*7$$!1,+Dcex diF*$\"1WG_6\"z]5)F*7$$!1++D1?QUcF*$\"1^OWZB'*\\%)F*7$$!1++D13%f+&F*$ \"1T7?'*e(Hx)F*7$$!1++D\"oS:P%F*$\"1\\@YB>gf!*F*7$$!1+++v@)*=PF*$\"1ov #eT#R;$*F*7$$!1++](G3U9$F*$\"1UoHilv4&*F*7$$!1*****\\-\\r\\#F*$\"1\"zI e;H)*o*F*7$$!1+++vGVZ=F*$\"1\"4ehRM)H)*F*7$$!1+++v4J@7F*$\"19%R?j7b#** F*7$$!1,+]iIKFl!#<$\"1#)eN*e/(y**F*7$$\"19+++DFOB!#=$\"1Vk;4F(*****F*7 $$\"1,+++!R5'fFhs$\"1yx\\n#QA)**F*7$$\"1++vV!QBE\"F*$\"1D(*Qy3V?**F*7$ $\"1******\\\"o?&=F*$\"1\"yoq'=)*G)*F*7$$\"1,+vVb4*\\#F*$\"1cG`VzM*o*F *7$$\"1,+DJ'=_6$F*$\"1\\,T(4#o=&*F*7$$\"1,+]P%y!ePF*$\"1n4TtU6-$*F*7$$ \"1,+v=WU[VF*$\"11VvwUOp!*F*7$$\"1++]7B>&)\\F*$\"1\"y%fy^\"Hy)F*7$$\"1 ++v$>:mk&F*$\"1&R!=*z(pZ%)F*7$$\"1++DcdQAiF*$\"1JuRtsvD\")F*7$$\"1,+]P PBWoF*$\"1`]\">MPyu(F*7$$\"1******\\Nm'[(F*$\"1>NM?G(fK(F*7$$\"1****\\ (yb^6)F*$\"152Ss#**R)oF*7$$\"1++vVVDB()F*$\"1X?i&3u/V'F*7$$\"1++]7TW)R *F*$\"1JBZ[m8**eF*7$$\"1+++:K^+5!#:$\"1rh+#z.()R&F*7$$\"1++]7,Hl5F_y$ \"12*f9//D%[F*7$$\"1+]P4w)R7\"F_y$\"19DKHu*3K%F*7$$\"1++]x%f\")=\"F_y$ \"1rLld#zOt$F*7$$\"1+]P/-a[7F_y$\"1DMA&\\tq;$F*7$$\"1+](=Yb;J\"F_y$\"1 Iw&e(>]iDF*7$$\"1++]i@Ot8F_y$\"1o\"[w*)R:'>F*7$$\"1+]PfL'zV\"F_y$\"1Rk *>\"oUC8F*7$$\"1+++!*>=+:F_y$\"1%>n%QmcbqFhs7$$\"1++DE&4Qc\"F_y$\"18f9 ,Pu')pF^t7$$\"1+]P%>5pi\"F_y$!11\\](RU%3cFhs7$$\"1+++bJ*[o\"F_y$!1:rkb V\\Q6F*7$$\"1++Dr\"[8v\"F_y$!1ynC[Zs&z\"F*7$$\"1+++Ijy5=F_y$!1uu[h\"Hp P#F*7$$\"1+]P/)fT(=F_y$!1Smf=yJ()HF*7$$\"1+]i0j\"[$>F_y$!1ZwZ@mLgNF*7$ $\"\"#Ffn$!1C9Zl$o9;%F*-FV6&FXFfnFfnFfn-F$6$7S7$F[oF[o7$F`oF`o7$FeoFeo 7$FjoFjo7$F_pF_p7$FdpFdp7$FipFip7$F^qF^q7$FcqFcq7$FhqFhq7$F]rF]r7$FbrF br7$FgrFgr7$F\\sF\\s7$FasFas7$FfsFfs7$F\\tF\\t7$FbtFbt7$FgtFgt7$F\\uF \\u7$FauFau7$FfuFfu7$F[vF[v7$F`vF`v7$FevFev7$FjvFjv7$F_wF_w7$FdwFdw7$F iwFiw7$F^xF^x7$FcxFcx7$FhxFhx7$F]yF]y7$FcyFcy7$FhyFhy7$F]zF]z7$FbzFbz7 $FgzFgz7$F\\[lF\\[l7$Fa[lFa[l7$Ff[lFf[l7$F[\\lF[\\l7$F`\\lF`\\l7$Fe\\l Fe\\l7$Fj\\lFj\\l7$F_]lF_]l7$Fd]lFd]l7$Fi]lFi]l7$F^^lF^^l-FV6&FXFfnFfn FY-%+AXESLABELSG6$%!GF]bl-%%VIEWG6$;F[oF^^l;F[o$\"\"\"Ffn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" }}}}{EXCHG {PARA 288 "" 0 "" {TEXT -1 45 "2. Find the periodic p oints of period 2 of " }{XPPEDIT 18 0 "y = 2*x*(1-x)" "6#/%\"yG*(\"\" #\"\"\"%\"xGF',&F'F'F(!\"\"F'" }{TEXT -1 77 ". . (Hint: the period po ints of order two of f would be the fixed points of " }{XPPEDIT 18 0 " f@@2" "6#-%#@@G6$%\"fG\"\"#" }{TEXT -1 87 " which are not fixed points of f. Classify them as repelling, attracting, or neither." }}} {EXCHG {PARA 288 "" 0 "" {TEXT -1 15 "Exercise: Let " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 114 " be a sequence of positive nu mbers converging to 0. Imagine yourself starting at the origin and tr avelling east " }{XPPEDIT 18 0 "a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 37 " miles, then turning north and going " }{XPPEDIT 18 0 "a[2]" "6#&%\"a G6#\"\"#" }{TEXT -1 18 " miles, then west " }{XPPEDIT 18 0 "a[3]" "6#& %\"aG6#\"\"$" }{TEXT -1 84 " miles, and so forth, cycling through the \+ directions as you go through the sequence " }{XPPEDIT 18 0 "a[n]" "6#& %\"aG6#%\"nG" }{TEXT -1 115 ". (1) Where do you end up? (2) How far do you travel along your path. Work the answers out for the sequen ces " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "1/2^n" "6#*&\"\"\"F$)\"\"#%\"nG!\"\"" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 9 "Solution:" }}{PARA 0 "" 0 " " {TEXT -1 51 "Call the point where we end up [x,y]. Then x = " } {XPPEDIT 18 0 "a1 - a3 + a5 - a7" "6#,*%#a1G\"\"\"%#a3G!\"\"%#a5GF%%#a 7GF'" }{TEXT -1 70 " ..., the sum of the alternating series of odd ter ms of the sequence " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 94 " and y is the sum of the alternating series of even terms of th e sequene. So for the sequence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a := n-> 1/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"n G6\"6$%)operatorG%&arrowGF(*&\"\"\"F-9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x = sum((-1)^n*a(2*n+1),n=0..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "y = sum((-1)^n*a(2*(n+1)),n=0..infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,$-%#lnG6#\"\"##\"\"\" F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "and for the sequence " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a := n-> 1/2^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\" \"F-)\"\"#9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " x = sum((-1)^n*a(2*n+1),n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG#\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " y = sum((-1)^n*a(2*(n+1)),n=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG#\"\"\"\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "(2) The total distance we travel along the path is the sum of a ll the distances travelled. So for the sequence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a := n->1/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"nG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-9$!\"\"F(F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "'distance' = sum(a(n),n=1.. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)distanceG%)infinityG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "the distance is infinity! \+ This is perhaps surprizing at first, since each time we turn we go a s maller distance than the last time. On the other hand, for the seque nce" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a := n->1/2^n;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"nG6\"6$%)operatorG%&arrowG F(*&\"\"\"F-)\"\"#9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "'distance' = sum(a(n),n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)distanceG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The distance travelled is only 1 unit." }}}{EXCHG {PARA 288 "" 0 " " {TEXT -1 165 "Exercise: Suppose we wanted to draw the paths describe d in the above problem. Here is a procedure which will do that. Use \+ it to draw the paths for the sequences " }{XPPEDIT 18 0 "an = 1/n" " 6#/%#anG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "an \+ = 1/2^n" "6#/%#anG*&\"\"\"F&)\"\"#%\"nG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 " cycle := proc(a,m) \nlocal path,i, dir,x,y,pt,ed; \nx := evalf(sum((-1 )^n*a(2*n+1),n=0..infinity));\ny := evalf(sum((-1)^n*a(2*(n+1)),n=0..i nfinity)); \n path := [0,0]; \n dir := 1,0; \n for i from 1 to m do \n path := path,[path[i][1]+dir[1]*a(i),\n path[i][2]+dir[2]*a(i)];\n \+ dir := -dir[2],dir[1]; \n od;\n pt := plot([path],scaling=constrained, color=red,thickness=2);\n ed := plot(\{[x,y]\},style=point,symbol=box) :\nplots[display]([pt,ed],title=cat(\"end at \",convert([x,y],string)) ); \n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 19 "For the sequenc 1/n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "cycle(n->1/n,15);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%727$\"\"!F(7$$\"\"\"F(F(7$F*$\" 1+++++++]!#;7$$\"1mmmmmmmmF/F-7$F1$\"1+++++++DF/7$$\"1nmmmmmm')F/F47$F 7$\"1nmmmmmmTF/7$$\"1Q_4Q_4QsF/F:7$F=$\"1nmmmmm;HF/7$$\"1]j?\\j?\\$)F/ F@7$FC$\"1nmmmmm;RF/7$$\"1Sa6Sa6SuF/FF7$FI$\"1LLLLLL$3$F/7$$\"14iM4iM4 #)F/FL7$FO$\"1Z!>w/>wz$F/7$$\"1U&zEazEa(F/FR-%'COLOURG6&%$RGBG$\"*++++ \"!\")F(F(-%*THICKNESSG6#\"\"#-F$6&7#7$$\"1+++N;)R&yF/$\"1+++.ftlMF/-F X6&FZ$\"#5!\"\"F(F(-%&STYLEG6#%&POINTG-%'SYMBOLG6#%$BOXG-%&TITLEG6#QBe nd~at~[.7853981635,~.3465735903]6\"-%+AXESLABELSG6$%!GFip-%(SCALINGG6# %,CONSTRAINEDG-%%VIEWG6$%(DEFAULTGFaq" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "cycle(n->1/2^n,15);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%727$\"\"!F(7$$\"1++++ +++]!#;F(7$F*$\"1+++++++DF,7$$\"1++++++]PF,F.7$F1$\"1++++++v=F,7$$\"1+ ++++]iSF,F47$F7$\"1+++++DJ?F,7$$\"1++++]P%)RF,F:7$F=$\"1++++v=#*>F,7$$ \"1+++]i!R+%F,F@7$FC$\"1+++DJ&>+#F,7$$\"1++]PM-**RF,FF7$FI$\"1++v=<^** >F,7$$\"1+]iSTC+SF,FL7$FO$\"1+DJq?7+?F,7$$\"1]P%['*Q***RF,FR-%'COLOURG 6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6&7#7$$\"1+++++++SF,$ \"1+++++++?F,-FX6&FZ$\"#5!\"\"F(F(-%&STYLEG6#%&POINTG-%'SYMBOLG6#%$BOX G-%&TITLEG6#QBend~at~[.4000000000,~.2000000000]6\"-%+AXESLABELSG6$%!GF ip-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$%(DEFAULTGFaq" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}}} {PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 7 3 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }