{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 " LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Luci da Console" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "Comic S ans MS" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Book Antiqua " 0 1 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 "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 "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 "Title " 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "Lucida Console" 1 14 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 "" 0 257 1 {CSTYLE "" -1 -1 "Lucida Console" 1 14 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 "" 0 258 1 {CSTYLE "" -1 -1 "Lucida Console " 1 14 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 "" 0 259 1 {CSTYLE "" -1 -1 "Lucida Console" 1 14 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 "" 0 260 1 {CSTYLE "" -1 -1 "Lucida Console" 1 14 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 {PARA 18 "" 0 "" {TEXT 257 19 "Mathematics I, 2001" }}{PARA 19 "" 0 "" {TEXT 258 10 "Yonggu Kim" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 256 "" 0 "" {TEXT -1 30 "Example 1.1.3) Find the limit " }{XPPEDIT 18 0 "Limit( n/(n+1),n = infinity);" "6#-%&LimitG6$*&%\"nG\"\"\",&F'F(\"\"\"F(!\"\" /F'%)infinityG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=n -> n/(n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\" nG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F-\"\"\"F0F0!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "points:=[seq([k,f(k)],k=1..4 0)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(\{points, 1\}, x=0.9..40,y=0.5..1.5, style=point, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7J7$$\"\"\"\"\"!$\"1+ ++++++]!#;7$$\"\"#F*$\"1mmmmmmmmF-7$$\"\"$F*$\"1+++++++vF-7$$\"\"%F*$ \"1+++++++!)F-7$$\"\"&F*$\"1MLLLLLL$)F-7$$\"\"'F*$\"1r&G9dG9d)F-7$$\" \"(F*$\"1++++++]()F-7$$\"\")F*$\"1))))))))))))))))F-7$$\"\"*F*$\"1++++ +++!*F-7$$\"#5F*$\"1\"4444444*F-7$$\"#6F*$\"1mmmmmmm\"*F-7$$\"#7F*$\"1 J#p2Bp2B*F-7$$\"#8F*$\"1'G9dG9dG*F-7$$\"#9F*$\"1LLLLLLL$*F-7$$\"#:F*$ \"1++++++v$*F-7$$\"#;F*$\"1`B)eqkF*$\"1+++++++&*F-7$$\"#?F*$\"1B&4Q_4Q_*F-7$$\"# @F*$\"1YXXXXXX&*F-7$$\"#AF*$\"1[VI\"R<_c*F-7$$\"#BF*$\"1MLLLLL$e*F-7$$ \"#CF*$\"1+++++++'*F-7$$\"#DF*$\"1;YQ:YQ:'*F-7$$\"#EF*$\"1H'H'H'H'H'*F -7$$\"#FF*$\"1Vr&G9dGk*F-7$$\"#GF*$\"1/Jz8Cun*F-7$$\"#JF*$\"1+++++](o*F-7$$\"#KF*$\"1(ppppp pp*F-7$$\"#LF*$\"1w6%HN#)eq*F-7$$\"#MF*$\"19dG9dG9(*F-7$$\"#NF*$\"1AAA AAAA(*F-7$$\"#OF*$\"1I(H(H(H(H(*F-7$$\"#PF*$\"1eJE0@%ot*F-7$$\"#QF*$\" 1V(*eV(*eV(*F-7$$\"#RF*$\"1++++++](*F-7$$\"#SF*$\"15c(4c(4c(*F--F$6#7S 7$FTF(7$$\"1nm\"zsoAv\"!#:F(7$$\"1L3-\">BQ\\#FbxF(7$$\"1n;aT(yxK$FbxF( 7$$\"1n;z!His;%FbxF(7$$\"1L3F@fv-]FbxF(7$$\"1nTNnaOxdFbxF(7$$\"1+v=&yG %zlFbxF(7$$\"1nT&e\"Q#*3uFbxF(7$$\"1+vVX'edB)FbxF(7$$\"1MLe)*)fi3*FbxF (7$$\"1o;Hl=QN)*FbxF(7$$\"1+]2x:(y1\"!#9F(7$$\"1+]i[7b_6FdzF(7$$\"1+]# H!e:M7FdzF(7$$\"1Zj\"FdzF(7$$\"1](oey[!>FdzF(7$$\"1n\"HZ1qI/ #FdzF(7$$\"1;ae8bFH@FdzF(7$$\"1]7B5wJ/AFdzF(7$$\"1nTXI^O&G#FdzF(7$$\"1 ++N\\^4pBFdzF(7$$\"1+vjP'35X#FdzF(7$$\"1](o%\\TEIDFdzF(7$$\"1+Dm#)QE=E FdzF(7$$\"1LL)obNtp#FdzF(7$$\"1+]iY9w\"y#FdzF(7$$\"1;a)3&QEeGFdzF(7$$ \"1+]n&=,>%HFdzF(7$$\"1$3-(*R(f?IFdzF(7$$\"1]Pk=w&G5$FdzF(7$$\"1L$e%=? 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\"\"#/&F%6#\"\"#-F-6#,&\"\"#\"\"\"&F%6#\"\"\"F?" }{TEXT -1 5 ",...," } {XPPEDIT 18 0 "x[n] = sqrt(2+x[n-1]);" "6#/&%\"xG6#%\"nG-%%sqrtG6#,&\" \"#\"\"\"&F%6#,&F'F-\"\"\"!\"\"F-" }{TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sqrt(2+sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&\"\"#\"\"\"*$-F%6#F(\" \"\"F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(2+%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&\"\"#\"\"\"*$-F%6#,&F(F) *$-F%6#F(\"\"\"F)F1F)F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h0dh>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sq:=proc(n::integer)\n option re member;\n sq(0):=0;\n sqrt(2+sq(n-1));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "sq(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$ -%%sqrtG6#,&\"\"#\"\"\"*$-F%6#,&F(F)*$-F%6#F(\"\"\"F)F1F)F1" }}} {EXCHG {PARA 0 "> " 0 "" 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{TEXT -1 51 "Using the Mathematical \+ Induction, we can show that " }{XPPEDIT 18 0 "x[n] <= 2;" "6#1&%\"xG6# %\"nG\"\"#" }{TEXT -1 9 " for all " }{XPPEDIT 18 0 "n;" "6#%\"nG" } {TEXT -1 28 " . Furthermore the sequence " }{XPPEDIT 18 0 "\{x[n]\};" "6#<#&%\"xG6#%\"nG" }{TEXT -1 83 " is an increasing sequence. So from \+ the completeness of real numbers, the sequence " }{TEXT 32 0 "" } {XPPEDIT 18 0 "\{x[n]\};" "6#<#&%\"xG6#%\"nG" }{TEXT -1 28 " converges to a real number " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 47 ", whic h is a least upper bound of the sequence " }{XPPEDIT 18 0 "\{x[n]\};" "6#<#&%\"xG6#%\"nG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 256 22 "Exercise 8) Show that " }{XPPEDIT 18 0 "Limi t(ln(n)/n,n = infinity) = 0;" "6#/-%&LimitG6$*&-%#lnG6#%\"nG\"\"\"F+! \"\"/F+%)infinityG\"\"!" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=n -> ln(n)/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGR6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%#lnG6#9$\"\"\"F0!\"\"F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "points:=[seq([k,f(k)],k =1..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(\{points , 0\}, x=10..100,y=0.0..0.3, style=point, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7`q7$$\"\"\"\"\" !F*7$$\"\"#F*$\"1E(*z-ftlM!#;7$$\"\"$F*$\"1KqAi4/iOF07$$\"\"%F*F.7$$\" \"&F*$\"1,#o[#e()=KF07$$\"\"'F*$\"1en/#yli)HF07$$\"\"(F*$\"1!f2:kr)zFF 07$$\"\")F*$\"1&z*4F>I*f#F07$$\"\"*F*$\"1)o%[T1OTCF07$$\"#5F*$\"1YS*H4 &e-BF07$$\"#6F*$\"1kIX$z/*z@F07$$\"#7F*$\"1++\\Tbvq?F07$$\"#8F*$\"10\" ys'z.t>F07$$\"#9F*$\"1%=D(\\4/&)=F07$$\"#:F*$\"12[t+oO0=F07$$\"#;F*$\" 1j)*R^z'Gt\"F07$$\"#F*$\"1uq(es/(\\:F07$$\"#?F*$\"1&*pxOh'y\\\"F07$$\"#@F*$\"11@R*es(\\9F 07$$\"#AF*$\"1`5)pH>]S\"F07$$\"#BF*$\"1/=va$eKO\"F07$$\"#CF*$\"16$yf4* =C8F07$$\"#DF*$\"1!GZ*H.b(G\"F07$$\"#EF*$\"1'=&3`S6`7F07$$\"#FF*$\"1WB u?.o?7F07$$\"#GF*$\"1(G[$RI2!>\"F07$$\"#HF*$\"1%y)4$\\O6;\"F07$$\"#IF* $\"1_SbgCtL6F07$$\"#JF*$\"1M%)*y!yt26F07$$\"#KF*$\"1:\\ipC/$3\"F07$$\" #LF*$\"1L*)*euZ&f5F07$$\"#MF*$\"1C7=g[;P5F07$$\"#NF*$\"1=T&=t8e,\"F07$ $\"#OF*$\"1%>*[t#>U&**!#<7$$\"#PF*$\"1Z6u,wBf(*Fcv7$$\"#QF*$\"1*pur:&f s&*Fcv7$$\"#RF*$\"1O))e5yu$R*Fcv7$$\"#SF*$\"1T[GN')>A#*Fcv7$$\"#TF*$\" 1j-iXG\\d!*Fcv7$$\"#UF*$\"1j^'oP8#**))Fcv7$$\"#VF*$\"1U3C8q(pu)Fcv7$$ \"#WF*$\"1my<')4V+')Fcv7$$\"#XF*$\"1xtEx*\\#f%)Fcv7$$\"#YF*$\"1n%G1ZLJ K)Fcv7$$\"#ZF*$\"1TP*yS.=>)Fcv7$$\"#[F*$\"11\"eg5-]1)Fcv7$$\"#\\F*$\"1 ,uI//\\UzFcv7$$\"#]F*$\"1#Hc3,YS#yFcv7$$\"#^F*$\"1e'\\\\-i%4xFcv7$$\"# _F*$\"1#fcEhX&)f(Fcv7$$\"#`F*$\"1rG-=o6\"\\(Fcv7$$\"#aF*$\"1Wvj$\\2qQ( Fcv7$$\"#bF*$\"16jyO.1'G(Fcv7$$\"#cF*$\"14*p\">!G\")=(Fcv7$$\"#dF*$\"1 O(4(*RsI4(Fcv7$$\"#eF*$\"1HpG6Qw+qFcv7$$\"#fF*$\"1C&*R8/36pFcv7$$\"#gF *$\"1,Nq.w!R#oFcv7$$\"#hF*$\"1I8D#[P\"RnFcv7$$\"#iF*$\"1af(HOoml'Fcv7$ $\"#jF*$\"13cgFVSwlFcv7$$\"#kF*$\"1([\\x\"[D)\\'Fcv7$$\"#lF*$\"1.zPhU8 AkFcv7$$\"#mF*$\"1pIMI<'zM'Fcv7$$\"#nF*$\"1lm&figcF'Fcv7$$\"#oF*$\"1j[ '**Qe^?'Fcv7$$\"#pF*$\"1`naQiQOhFcv7$$\"#qF*$\"1qL@<*y#pgFcv7$$\"#rF*$ \"1zgquWx.gFcv7$$\"#sF*$\"1l*)=aS\")RfFcv7$$\"#tF*$\"1vK?+yu_Fcv7$ $\"#&)F*$\"1Ju/P&[mA&Fcv7$$\"#')F*$\"1wZH+OZz^Fcv7$$\"#()F*$\"1v9fAxAL ^Fcv7$$\"#))F*$\"1N_sVF)y3&Fcv7$$\"#*)F*$\"1FWQ^;TV]Fcv7$$\"#!*F*$\"1% =*eA&)y**\\Fcv7$$\"#\"*F*$\"1O)*po%))p&\\Fcv7$$\"##*F*$\"1d*[Pe()\\\" \\Fcv7$$\"#$*F*$\"1;ZFcv7$$\"# )*F*$\"1+9aV#Q&yYFcv7$$\"#**F*$\"1_h:-_`TYFcv7$$\"$+\"F*$\"1#4))f=q^g% Fcv-F$6#7S7$FSF*7$$\"1++](Quh>\"!#9F*7$$\"1+]7tY'oO\"F[[mF*7$$\"1++D^P #)e:F[[mF*7$$\"1++ve_0_F[[mF*7$$\"1+]7VsmA@F[[mF *7$$\"1+]7)R&G2BF[[mF*7$$\"1+]7ex@)\\#F[[mF*7$$\"1+]i&zP&)o#F[[mF*7$$ \"1++]Z`I%)GF[[mF*7$$\"1++v8vtcIF[[mF*7$$\"1++]#Hb3D$F[[mF*7$$\"1++]P, xXMF[[mF*7$$\"1++]2ngLOF[[mF*7$$\"1**\\73.=/QF[[mF*7$$\"1++]<)3q+%F[[m F*7$$\"1+++q6$)yTF[[mF*7$$\"1+]789qyVF[[mF*7$$\"1+++X/ibXF[[mF*7$$\"1+ ]7j'G(\\ZF[[mF*7$$\"1+]P*elX$\\F[[mF*7$$\"1++DJNUF^F[[mF*7$$\"1,]iDt_/ `F[[mF*7$$\"1++v$pdb\\&F[[mF*7$$\"1+]7eX)Rp&F[[mF*7$$\"1+](os:n'eF[[mF *7$$\"1++D@,F`gF[[mF*7$$\"1*****\\1**fC'F[[mF*7$$\"1***\\itYXV'F[[mF*7 $$\"1+]7.j(ph'F[[mF*7$$\"1***\\PBL&>oF[[mF*7$$\"1+++X'R:+(F[[mF*7$$\"1 ++]P.(e>(F[[mF*7$$\"1+]7GG'>P(F[[mF*7$$\"1++]K%yWc(F[[mF*7$$\"1+]781iX xF[[mF*7$$\"1+]i&Qm\\$zF[[mF*7$$\"1****\\(['3?\")F[[mF*7$$\"1+]7y+*QJ) F[[mF*7$$\"1+++qfa+&)F[[mF*7$$\"1,+vy&G9p)F[[mF*7$$\"1,]7$eI2)))F[[mF* 7$$\"1,++l%zY0*F[[mF*7$$\"1++v8X/a#*F[[mF*7$$\"1+++!**eBV*F[[mF*7$$\"1 ,]78%zCi*F[[mF*7$$\"1+](o\"*[W!)*F[[mF*7$F`jlF*-%&STYLEG6#%&POINTG-%+A XESLABELSG6$Q\"x6\"Q\"yF_dm-%'COLOURG6&%$RGBGF*F*$\"*++++\"!\")-%%VIEW G6$;FSF`jl;F*$F3!\"\"" 1 5 0 1 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Limit(f(n),n =infinity)= limit(f(n),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%&LimitG6$*&-%#lnG6#%\"nG\"\"\"F+!\"\"/F+%)infinityG\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }