{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 257 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 257 "" 0 "" {TEXT -1 35 "Lesson 23: Power Series Expansions" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 136 "In this lesson, \+ we explore methods of expanding functions into power series. The basic idea hinges on the geometric series expansion of " }{XPPEDIT 18 0 "1/ (1-x)" "6#*&\"\"\"F$,&F$F$%\"xG!\"\"F'" }{TEXT -1 407 ". However, usin g differentiation and integration we can expand many more functions in to power series also. In addition, we will examine the interval of con vergence and how it is affected by the location of the expansion and f eatures of the function such as vertical asymptotes. In general, this \+ module will reinforce methods one might use by hand and not rely on th e automated expansions Maple can generate." }}{PARA 5 "" 0 "" {TEXT -1 83 "_______________________________________________________________ ____________________" }}{PARA 4 "" 0 "" {TEXT -1 33 "A. Expand A Funct ion As A Series" }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________ ______________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "An infini te geometric series can be simplified." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Sum( a*r^k, k = 0..infin ity): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&%\" aG\"\"\")%\"rG%\"kGF)/F,;\"\"!%)infinityG,$*&F(F),&F+F)F)!\"\"F4F4" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We can t urn this process around. Staring with an expression, we can expand it \+ into a series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a/(1 - r): % = series( %, r);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&%\"aG\"\"\",&F&F&%\"rG!\"\"F)+1F(F%\"\"!F%F&F%\"\" #F%\"\"$F%\"\"%F%\"\"&-%\"OG6#F&\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "Given an expression in x, we can comp ute the series expansion. You can also do these by hand." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "serie s( 1/(1-x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!F %F%F%\"\"#F%\"\"$F%\"\"%F%\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "series( 1/(1-x), x, 15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+C%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$F%\"\"%F%\"\"&F%\" \"'F%\"\"(F%\"\")F%\"\"*F%\"#5F%\"#6F%\"#7F%\"#8F%\"#9-%\"OG6#F%\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 205 "Maple will expand to a certain number of terms as a default. We can also sp ecify to what power of x we want to expand. Maple then uses the \"big \+ Oh\" notation to indicate the order of the error or remainder." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "All of t he previous examples expanded the power series about x = 0. We can als o expand about other values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "series( 1/(1-x), x = 3, 12); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+=,&%\"xG\"\"\"\"\"$!\"\"#F(\"\"# \"\"!#F&\"\"%F&#F(\"\")F*#F&\"#;F'#F(\"#KF-#F&\"#k\"\"&#F(\"$G\"\"\"'# F&\"$c#\"\"(#F(\"$7&F/#F&\"%C5\"\"*#F(\"%[?\"#5#F&\"%'4%\"#6-%\"OG6#F& \"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "series( 1/(2 + x^2) , x = -1,12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+=,&%\"xG\"\"\"F&F&#F &\"\"$\"\"!#\"\"#\"\"*F&#F&\"#FF+#!\"%\"#\")F(#!#6\"$V#\"\"%#!#5\"$H( \"\"&#\"#8\"%(=#\"\"'#\"#c\"%hl\"\"(#\"#t\"&$o>\"\")#!#A\"&\\!fF,#!$j# \"'Zr<\"#5#!$g%\"'T9`\"#6-%\"OG6#F&\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 83 "__________________________________ _________________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 37 "B. Integrate / Series / Differentiate" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________________________ __________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 331 "The method we used above to expand a series into a geometric series works only in certain cases. If an expression does not lend itself readily to this method, there are other tricks. One i s to intergrate the function, expand the anti-derivative into a series , then differentiate the result. In this indirect way we find the seri es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Le ts consider an example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "We start with a function, integrate it, expand it i nto a power series, then differentiate to get back to the original fun ction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> 3/(x-5)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.*$),&9$F.\"\"&! \"\"\"\"#F.F4\"\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F := int( f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,$*&\"\" \"F',&%\"xGF'\"\"&!\"\"F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series( %, x, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"xG#\" \"$\"\"&\"\"!#F&\"#D\"\"\"#F&\"$D\"\"\"##F&\"$D'F&#F&\"%DJ\"\"%#F&\"&D c\"F'#F&\"&D\"y\"\"'#F&\"'D1R\"\"(#F&\"(DJ&>\"\")#F&\"(Dcw*\"\"*-%\"OG 6#F+\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff( %, x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG#\"\"$\"#D\"\"!#\"\"'\"$D\"\" \"\"#\"\"*\"$D'\"\"##\"#7\"%DJF&#F&F3\"\"%#\"#=\"&D\"y\"\"&#\"#@\"'D1R F*#\"#C\"(DJ&>\"\"(#\"#F\"(Dcw*\"\")-%\"OG6#F,F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 226 "We can convert the new f ound series into a polynomial and evaluate it to demonstrate that it t ake values close to the original function. The reason for the slight e rror is that we are only taking a finite number of terms here." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g := unapply(convert( %, polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,4#\"\"$\"#D\" \"\"*&#\"\"'\"$D\"F09$F0F0*&#\"\"*\"$D'F0)F5\"\"#F0F0*&#\"#7\"%DJF0)F5 F.F0F0*&#F.F?F0)F5\"\"%F0F0*&#\"#=\"&D\"yF0)F5\"\"&F0F0*&#\"#@\"'D1RF0 )F5F3F0F0*&#\"#C\"(DJ&>F0)F5\"\"(F0F0*&#\"#F\"(Dcw*F0)F5\"\")F0F0F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f(1); g(1); f(1) - g(1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"(Z5$=\"(Dcw*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$ B\"\"*++Dc\"" }}}{PARA 5 "" 0 "" {TEXT -1 83 "________________________ ___________________________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 37 "C. Differentiate / Series / Integrate" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________________________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "We can use the same trick in reverse t oo. Take a function, differentiate it, expand into a power series, and integrate. When we integrate, the constant of integration will become the constant term of the series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 25 "Lets consider an example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "We start with a fun citon, integrate it, expand it into a power series, then differentiate to get back to the original function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> arctan(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%'arctanG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "diff( f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&F$F$*$)%\"xG\"\"#F$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "series( %, x, 14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"\"\"!!\"\"\"\"#F%\"\"%F'\"\"'F%\"\")F'\"#5F%\"#7-% \"OG6#F%\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Int( %, x) \+ : % = int( %%, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$+3%\"x G\"\"\"\"\"!!\"\"\"\"#F)\"\"%F+\"\"'F)\"\")F+\"#5F)\"#7-%\"OG6#F)\"#9F (+3F(F)F)#F+\"\"$F8#F)\"\"&F:#F+\"\"(F<#F)\"\"*F>#F+\"#6F@#F)\"#8FBF2 \"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 392 "Every indefinite integral has a constant of integration. In particula r, when we integrate the series, we can't disregard the constant of in tegration. It turns out to be constant term of the series. Lets look a t an example. We start with a function , and compute its series expans ion by the method described above. Then we convert the series to a pol ynomial so we can graph it along with f(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := x -> 1/(1-x); c := -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)op eratorG%&arrowGF(*&\"\"\"F-,&F-F-9$!\"\"F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "series( D(f)(x), x=c, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1,&% \"xG\"\"\"F&F&#F&\"\"%\"\"!F'F&#\"\"$\"#;\"\"##F&\"\")F+#\"\"&\"#kF(#F +F2F1-%\"OG6#F&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g : = unapply( convert( int(%, x), polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,09$#\"\"\"\"\" %F.F/*&#F/\"\")F/),&F-F/F/F/\"\"#F/F/*&#F/\"#;F/)F5\"\"$F/F/*&#F/\"#KF /)F5F0F/F/*&#F/\"#kF/)F5\"\"&F/F/*&#F/\"$G\"F/)F5\"\"'F/F/F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot( [f(x), g(x)], x = -5.. 1, y = -2..5, thickness=[3,2], color=[red,coral]);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7ep7$$!\"&\"\"!$\"3 emmmmmmm;!#=7$$!3!******\\2<#p[!#<$\"3q^zshZ!Qq\"F-7$$!3!)*****4bBav%F 1$\"3)fi*3(e\"\\P)[5Q^?'=F-7$$!3q*****4/>F-7$$!3N+++,kZGTF1$\"3S))f(R&o*)\\>F-7$$!3(******4;)=,SF1 $\"3Y$H6&[[_**>F-7$$!3y*****f83V(QF1$\"3!RnNA@t:0#F-7$$!3:+++NkzVPF1$ \"3*RdrfD;!3@F-7$$!3\")*****zlT)GOF1$\"3o9li)pn.;#F-7$$!37+++0)H%*\\$F 1$\"3hTKoNQ]AAF-7$$!3#)*****\\d'[pLF1$\"3-&)yd.%)f)G#F-7$$!38+++&>iUC$ F1$\"3[YgS5@7cBF-7$$!3y*****4YY08$F1$\"3p\\h5tr)4U#F-7$$!3)******\\XF` *HF1$\"3!\\LqfvBH]#F-7$$!3)*******>#z2)GF1$\"37O5t)3-od#F-7$$!3-+++\"R Kvu#F1$\"3OEuFmDUoEF-7$$!3<+++qjeHEF1$\"3qEq!y([8bFF-7$$!3&)*****4*3=+ DF1$\"3Cj=l)>&*p&GF-7$$!3')*****RFcpP#F1$\"3E#3%o:lChHF-7$$!3y*****>J% Q[AF1$\"3)z-&=$[`%yIF-7$$!3z*****f6:.8#F1$\"3a0!3EcmX>$F-7$$!35+++Q:'H +#F1$\"3Dv?w!)f/ILF-7$$!3-+++hpnq=F1$\"35UY\\()))\\$[$F-7$$!37+++\\G_b %)opP\"F1$\"3OI/8R(Qq?%F-7$$!3-+++J\"\\ `D\"F1$\"3D5I8?L!RV%F-7$$!3\"******z<6.7\"F1$\"3w$G2a'))G;ZF-7$$!3y*** ****pN(*)**F-$\"3J1Wi#RnD+&F-7$$!3/+++]x>%p)F-$\"3%p#HA`LD\\`F-7$$!3J+ ++5yC?vF-$\"3tFjID=o2dF-7$$!3!*******\\/\"oB'F-$\"3y#[h/2X)ehF-7$$!3b+ ++5f>H]F-$\"3o!3MR\"fr`mF-7$$!37+++g2*ow$F-$\"3u'***>yU!QE(F-7$$!3y*** ***\\nvKDF-$\"3Q7'[UZ!4zzF-7$$!3%*******4GtS7F-$\"3aEHu\\r@'*))F-7$$\" 3B+++++!)RO!#@$\"3]y'HD6k.+\"F17$$\"35+++?0>w7F-$\"3;cMO:#)GY6F17$$\"3 !)*******)Q?QDF-$\"356!p&4*f,M\"F17$$\"31++++J'yp$F-$\"3+VxFsLw'e\"F17 $$\"3=+++?M'p-&F-$\"3Jey*[&R3**GN F17$$\"3U*******3'>$[(F-$\"3+4^4bKHtRF17$$\"3Q+++&fyky(F-$\"3mz4^6%)o< XF17$$\"3E++++6w*3)F-$\"3(Q)\\q9t%\\B&F17$$\"3;+++_BST#)F-$\"3/tw,8!\\ jo&F17$$\"37+++0O/$R)F-$\"3$oSTP@WHA'F17$$\"3a+++IU')o%)F-$\"3ZOhef%*4 JlF17$$\"3-+++d[oW&)F-$\"3)*)osM%ROroF17$$\"3_+++$[00i)F-$\"3Qn3Eu?.\\ sF17$$\"3)*******4hK'p)F-$\"3#)4V*4'*H1n(F17$$\"3++++Hd!yx)F-$\"3=q\\e (H0?=)F17$$\"3#******pM&Gf))F-$\"3%38+pCMkw)F17$$\"3'******p:D+!*)F-$ \"3%)*)\\v/q6\"4*F17$$\"3#******f'\\wS*)F-$\"3h><4^`xS%*F17$$\"3))**** *\\x/:)*)F-$\"36)QN^O1%=)*F17$$\"3w*****HeWA-*F-$\"36<*))>m]F-\"!#;7$$ \"3!)*****HR%)H1*F-$\"3#)*)o;$3=s1\"Fj_l7$$\"3x*****>?CP5*F-$\"387\"Qg *zs:6Fj_l7$$\"3t*****4,kW9*F-$\"3[b8h+z&)o6Fj_l7$$\"3o******>Q?&=*F-$ \"3;B>k&z+tA\"Fj_l7$$\"3l******GO%fA*F-$\"3)*=*fVd&*=H\"Fj_l7$$\"3g*** **zV$om#*F-$\"3oZKr]vmj8Fj_l7$$\"3d*****pCBuI*F-$\"3V(\\!**)G$)QW\"Fj_ l7$$\"3c+++bI;[$*F-$\"3@j))>#*f7M:Fj_l7$$\"3[*****\\'G!*)Q*F-$\"3s'*e& 31,kj\"Fj_l7$$\"3c+++uEkH%*F-$\"3GNq\"4&oG`g#f*F-$\"3_\\QG\"f,YX#Fj_l7$$\"3O+++>\"[5*)>DoIFj_l7$$\"3G+ ++P8#[r*F-$\"3`qS$=qtl]$Fj_l7$$\"3C+++Y6cb(*F-$\"33\\(R@l-54%Fj_l7$$\" 3?+++b4I'z*F-$\"3.*pnD=.#4\\Fj_l7$$\"3;+++k2/P)*F-$\"3(Ri4#yR]OhFj_l7$ $\"37+++t0yx)*F-$\"3;)\\zUI0?=)Fj_l7$$\"33+++#Q?&=**F-$\"3zC>k&z+tA\"! #:7$$\"3/+++\">g#f**F-$\"3e\\QG\"f,YX#Fifl7$%%FAILGF`gl-%'COLOURG6&%$R GBG$\"*++++\"!\")$F*F*Fhgl-%*THICKNESSG6#\"\"$-F$6%7Z7$F($\"#@F*7$$!3U ++voUIn\\F1$\"3kmx0di,#*>Fj_l7$$!3&*****\\P&3Y$\\F1$\"3oI&H*3Cl))=Fj_l 7$$!3[***\\i!G\">!\\F1$\"3S@6>d%[(*y\"Fj_l7$F/$\"3ek&H>i[^p\"Fj_l7$$!3 U**\\78.K7[F1$\"3`D#e1[S.a\"Fj_l7$$!3#)***\\7bBav%F1$\"3=v[p&HRtR\"Fj_ l7$$!3'***\\(=>P9p%F1$\"3/M#p%fKr\\7Fj_l7$$!36++]K3XFYF1$\"3HY)\\5#>5: 6Fj_l7$$!3w******H./jXF1$\"34A=j TF17$$!3*****\\7;)=,SF1$\"32:)>O)3\"e7$F17$$!3!)***\\i83V(QF1$\"3]M!z0 )[)HI#F17$Fgn$\"3W3%4[;\"*)Q;F17$$!3w****\\d;%)GOF1$\"3E:(et0n/=\"F17$ Fao$\"3#*)Q6`y>L!yF-7$Ffo$\"3%4)R,&)R\\'y%F-7$F[p$\"3')y'HZTIXj#F-7$$! 3!)***\\7YY08$F1$\"3g?c\"RII**=\"F-7$Fep$!3qwZX6(H*zM!#?7$Fjp$!3Wj*=[1 `MZ(!#>7$$!3/++D\"RKvu#F1$!3KJ/:!>)*RH\"F-7$Fdq$!3)[`/YX-!)e\"F-7$$!3( )***\\7*3=+DF1$!39r4rI*>8w\"F-7$$!3%)***\\PFcpP#F1$!3E4?1\"eI;#=F-7$$! 3#)****\\7VQ[AF1$!3?(3[Hz)*y!=F-7$$!3\")***\\i6:.8#F1$!3C1/#[X.mu\"F-7 $$!31++]P:'H+#F1$!3Z'z$z()QRV;F-7$$!3/++Dhpnq=F1$!34BC.L\"yh]\"F-7$$!3 5++v[G_b\"F-7$$!3@+ ++!HnE]\"F1$!3@EC9\")*4S+\"F-7$$!3=++]U)opP\"F1$!3K#*)GU1d#HzF^_m7$$!3 /++DJ\"\\`D\"F1$!31DPy$GU4m&F^_m7$$!34++]x6J?6F1$!3`9VO+76PGF^_m7$$!3A /+++dt*)**F-$\"3GzYSCERnDF_x7$$!3%*)*****\\x>%p)F-$\"3EhKphK`#\\$F^_m7 $$!3u-+]7yC?vF-$\"3E9CeMDzwqF^_m7$$!3c'*****\\/\"oB'F-$\"3[MNbWOze6F-7 $$!33/+]7f>H]F-$\"3wb^u8hK`;F-7$$!3A****\\i2*ow$F-$\"3?)3*e,*Hw7F-$\"3'os'>T`GbiF-7$$\"3Z.+]( )Q?QDF-$\"3=KMTw6d\"*yF-7$$\"3%G+++5jyp$F-$\"3w;T-\\g%fu*F-7$$\"3r,++D M'p-&F-$\"3`(\\Rh(G.R7F17$F\\z$\"3?$f#H-D(Q`\"F17$$\"3U,+](3'>$[(F-$\" 3WvDoMZXB>F17$$\"3I,+]7hK'p)F-$\"3S\"HxZ)eE&Q#F17$$\"\"\"F*$F\\hlF*-Fb gl6&FdglFegl$\")AR!)\\FgglFhgl-Fjgl6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yF `im-%%VIEWG6$;F(Fbhm;$!\"#F*$\"\"&F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 216 "The thick red curve is the ori ginal function, and the thinner yellow curve is our series - which is \+ supposed to represent our function. However, as you can see, it doesn' t look like it's too good of a representation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "The one thing we forgot \+ was to add the constant term, which is f(-1). We will now define a fun ction h(x) = g(x) + f(-1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h := unapply( g(x) + f(c), x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowG F(,09$#\"\"\"\"\"%#\"\"$F0F/*&#F/\"\")F/),&F-F/F/F/\"\"#F/F/*&#F/\"#;F /)F7F2F/F/*&#F/\"#KF/)F7F0F/F/*&#F/\"#kF/)F7\"\"&F/F/*&#F/\"$G\"F/)F7 \"\"'F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot( [ f(x), g(x), h(x), [[c,0],[c,f(c)]] ], x= -5..1, y = -2..5, thickness= \+ [3,1,2,2], color = [red, coral, blue, green]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7ep7$$!\"&\"\"!$\"3em mmmmmm;!#=7$$!3!******\\2<#p[!#<$\"3q^zshZ!Qq\"F-7$$!3!)*****4bBav%F1$ \"3)fi*3(e\"\\P)[5Q^?'=F-7$$!3q*****4/>F-7$$!3N+++,kZGTF1$\"3S))f(R&o*)\\>F-7$$!3(******4;)=,SF1$ \"3Y$H6&[[_**>F-7$$!3y*****f83V(QF1$\"3!RnNA@t:0#F-7$$!3:+++NkzVPF1$\" 3*RdrfD;!3@F-7$$!3\")*****zlT)GOF1$\"3o9li)pn.;#F-7$$!37+++0)H%*\\$F1$ \"3hTKoNQ]AAF-7$$!3#)*****\\d'[pLF1$\"3-&)yd.%)f)G#F-7$$!38+++&>iUC$F1 $\"3[YgS5@7cBF-7$$!3y*****4YY08$F1$\"3p\\h5tr)4U#F-7$$!3)******\\XF`*H F1$\"3!\\LqfvBH]#F-7$$!3)*******>#z2)GF1$\"37O5t)3-od#F-7$$!3-+++\"RKv u#F1$\"3OEuFmDUoEF-7$$!3<+++qjeHEF1$\"3qEq!y([8bFF-7$$!3&)*****4*3=+DF 1$\"3Cj=l)>&*p&GF-7$$!3')*****RFcpP#F1$\"3E#3%o:lChHF-7$$!3y*****>J%Q[ AF1$\"3)z-&=$[`%yIF-7$$!3z*****f6:.8#F1$\"3a0!3EcmX>$F-7$$!35+++Q:'H+# F1$\"3Dv?w!)f/ILF-7$$!3-+++hpnq=F1$\"35UY\\()))\\$[$F-7$$!37+++\\G_b%)opP\"F1$\"3OI/8R(Qq?%F-7$$!3-+++J\"\\`D \"F1$\"3D5I8?L!RV%F-7$$!3\"******z<6.7\"F1$\"3w$G2a'))G;ZF-7$$!3y***** **pN(*)**F-$\"3J1Wi#RnD+&F-7$$!3/+++]x>%p)F-$\"3%p#HA`LD\\`F-7$$!3J+++ 5yC?vF-$\"3tFjID=o2dF-7$$!3!*******\\/\"oB'F-$\"3y#[h/2X)ehF-7$$!3b+++ 5f>H]F-$\"3o!3MR\"fr`mF-7$$!37+++g2*ow$F-$\"3u'***>yU!QE(F-7$$!3y***** *\\nvKDF-$\"3Q7'[UZ!4zzF-7$$!3%*******4GtS7F-$\"3aEHu\\r@'*))F-7$$\"3B +++++!)RO!#@$\"3]y'HD6k.+\"F17$$\"35+++?0>w7F-$\"3;cMO:#)GY6F17$$\"3!) *******)Q?QDF-$\"356!p&4*f,M\"F17$$\"31++++J'yp$F-$\"3+VxFsLw'e\"F17$$ \"3=+++?M'p-&F-$\"3Jey*[&R3**GNF1 7$$\"3U*******3'>$[(F-$\"3+4^4bKHtRF17$$\"3Q+++&fyky(F-$\"3mz4^6%)o<4^`xS%*F17$$\"3))*****\\ x/:)*)F-$\"36)QN^O1%=)*F17$$\"3w*****HeWA-*F-$\"36<*))>m]F-\"!#;7$$\"3 !)*****HR%)H1*F-$\"3#)*)o;$3=s1\"Fj_l7$$\"3x*****>?CP5*F-$\"387\"Qg*zs :6Fj_l7$$\"3t*****4,kW9*F-$\"3[b8h+z&)o6Fj_l7$$\"3o******>Q?&=*F-$\"3; B>k&z+tA\"Fj_l7$$\"3l******GO%fA*F-$\"3)*=*fVd&*=H\"Fj_l7$$\"3g*****zV $om#*F-$\"3oZKr]vmj8Fj_l7$$\"3d*****pCBuI*F-$\"3V(\\!**)G$)QW\"Fj_l7$$ \"3c+++bI;[$*F-$\"3@j))>#*f7M:Fj_l7$$\"3[*****\\'G!*)Q*F-$\"3s'*e&31,k j\"Fj_l7$$\"3c+++uEkH%*F-$\"3GNq\"4&oG`g#f*F-$\"3_\\QG\"f,YX#Fj_l7$$\"3O+++>\"[5*)>DoIFj_l7$$\"3G+++P8 #[r*F-$\"3`qS$=qtl]$Fj_l7$$\"3C+++Y6cb(*F-$\"33\\(R@l-54%Fj_l7$$\"3?++ +b4I'z*F-$\"3.*pnD=.#4\\Fj_l7$$\"3;+++k2/P)*F-$\"3(Ri4#yR]OhFj_l7$$\"3 7+++t0yx)*F-$\"3;)\\zUI0?=)Fj_l7$$\"33+++#Q?&=**F-$\"3zC>k&z+tA\"!#:7$ $\"3/+++\">g#f**F-$\"3e\\QG\"f,YX#Fifl7$%%FAILGF`gl-%'COLOURG6&%$RGBG$ \"*++++\"!\")$F*F*Fhgl-%*THICKNESSG6#\"\"$-F$6%7Z7$F($\"#@F*7$$!3U++vo UIn\\F1$\"3kmx0di,#*>Fj_l7$$!3&*****\\P&3Y$\\F1$\"3oI&H*3Cl))=Fj_l7$$! 3[***\\i!G\">!\\F1$\"3S@6>d%[(*y\"Fj_l7$F/$\"3ek&H>i[^p\"Fj_l7$$!3U** \\78.K7[F1$\"3`D#e1[S.a\"Fj_l7$$!3#)***\\7bBav%F1$\"3=v[p&HRtR\"Fj_l7$ $!3'***\\(=>P9p%F1$\"3/M#p%fKr\\7Fj_l7$$!36++]K3XFYF1$\"3HY)\\5#>5:6Fj _l7$$!3w******H./jXF1$\"34A=jTF1 7$$!3*****\\7;)=,SF1$\"32:)>O)3\"e7$F17$$!3!)***\\i83V(QF1$\"3]M!z0)[) HI#F17$Fgn$\"3W3%4[;\"*)Q;F17$$!3w****\\d;%)GOF1$\"3E:(et0n/=\"F17$Fao $\"3#*)Q6`y>L!yF-7$Ffo$\"3%4)R,&)R\\'y%F-7$F[p$\"3')y'HZTIXj#F-7$$!3!) ***\\7YY08$F1$\"3g?c\"RII**=\"F-7$Fep$!3qwZX6(H*zM!#?7$Fjp$!3Wj*=[1`MZ (!#>7$$!3/++D\"RKvu#F1$!3KJ/:!>)*RH\"F-7$Fdq$!3)[`/YX-!)e\"F-7$$!3()** *\\7*3=+DF1$!39r4rI*>8w\"F-7$$!3%)***\\PFcpP#F1$!3E4?1\"eI;#=F-7$$!3#) ****\\7VQ[AF1$!3?(3[Hz)*y!=F-7$$!3\")***\\i6:.8#F1$!3C1/#[X.mu\"F-7$$! 31++]P:'H+#F1$!3Z'z$z()QRV;F-7$$!3/++Dhpnq=F1$!34BC.L\"yh]\"F-7$$!35++ v[G_b\"F-7$$!3@+++! HnE]\"F1$!3@EC9\")*4S+\"F-7$$!3=++]U)opP\"F1$!3K#*)GU1d#HzF^_m7$$!3/++ DJ\"\\`D\"F1$!31DPy$GU4m&F^_m7$$!34++]x6J?6F1$!3`9VO+76PGF^_m7$$!3A/++ +dt*)**F-$\"3GzYSCERnDF_x7$$!3%*)*****\\x>%p)F-$\"3EhKphK`#\\$F^_m7$$! 3u-+]7yC?vF-$\"3E9CeMDzwqF^_m7$$!3c'*****\\/\"oB'F-$\"3[MNbWOze6F-7$$! 33/+]7f>H]F-$\"3wb^u8hK`;F-7$$!3A****\\i2*ow$F-$\"3?)3*e,*Hw7F-$\"3'os'>T`GbiF-7$$\"3Z.+]()Q? QDF-$\"3=KMTw6d\"*yF-7$$\"3%G+++5jyp$F-$\"3w;T-\\g%fu*F-7$$\"3r,++DM'p -&F-$\"3`(\\Rh(G.R7F17$F\\z$\"3?$f#H-D(Q`\"F17$$\"3U,+](3'>$[(F-$\"3Wv DoMZXB>F17$$\"3I,+]7hK'p)F-$\"3S\"HxZ)eE&Q#F17$$\"\"\"F*$F\\hlF*-Fbgl6 &FdglFegl$\")AR!)\\FgglFhgl-Fjgl6#Fchm-F$6%7Z7$F($\"3+++++++]@Fj_l7$Fd hl$\"3kmx0di,U?Fj_l7$Fihl$\"3oI&H*3ClQ>Fj_l7$F^il$\"3R@6>d%[(R=Fj_l7$F /$\"3ek&H>i[^u\"Fj_l7$Ffil$\"3aD#e1[S.f\"Fj_l7$F[jl$\"3=v[p&HRtW\"Fj_l 7$F`jl$\"3/M#p%fKr*H\"Fj_l7$Fejl$\"3IY)\\5#>5l6Fj_l7$Fjjl$\"3@s^2?VzT5 Fj_l7$F_[m$\"3!eSR-MC#)H*F17$Fd[m$\"3iEz3To$zG)F17$Fi[m$\"3Ki-O')))ett F17$F^\\m$\"3j+cXQ&[q!fF17$Fc\\m$\"3UhH**\\>=jYF17$Fh\\m$\"32:)>O)3\"e i$F17$F]]m$\"3]M!z0)[)H!GF17$Fgn$\"3W3%4[;\"*)Q@F17$Fe]m$\"3E:(et0n/o \"F17$Fao$\"3!*Q6`y>L!G\"F17$Ffo$\"3%4)R,&)R\\'y*F-7$F[p$\"3')y'HZTIXj (F-7$Fc^m$\"3g?c\"RII**='F-7$Fep$\"3C_a)Gq+_'\\F-7$Fjp$\"3m.\"=NpaED%F -7$F`_m$\"3So&\\)4=+1PF-7$Fdq$\"3SlaRXv*>T$F-7$Fh_m$\"3&)G!*Gp+oQKF-7$ F]`m$\"3u!*z$*=%p$yJF-7$Fb`m$\"3#R\">0275#>$F-7$Fg`m$\"3w$fz^a'R`KF-7$ F\\am$\"3)H?1A61mN$F-7$Faam$\"3Yxv'p'=#Q\\$F-7$Ffam$\"3UCa:K$fIj$F-7$F [bm$\"3hrVHY%*z,QF-7$F`bm$\"3asv&)=+*f*RF-7$Febm$\"306rd$Huq?%F-7$Fjbm $\"3OF;ird!RV%F-7$F_cm$\"3FoN'*z))G;ZF-7$Fdcm$\"34/Wi#RnD+&F-7$Ficm$\" 3[E$phK`#\\`F-7$F^dm$\"3,T#eMDzwq&F-7$Fcdm$\"31MNbWOzehF-7$Fhdm$\"3#\\ :XP6ELl'F-7$F]em$\"3%z3*e,*HT`Gb7\"F17$Fffm$\"3+V8k " 0 "" {MPLTEXT 1 0 18 "f := x -> ln (1+x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operato rG%&arrowGF(-%#lnG6#,&9$\"\"\"F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff( f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& \"\"\"F$,&%\"xGF$F$F$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series( %, x, 7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\" \"\"!!\"\"F%F%\"\"#F'\"\"$F%\"\"%F'\"\"&F%\"\"'-%\"OG6#F%\"\"(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g := unapply( int(convert( % , polynom), x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6 \"6$%)operatorG%&arrowGF(,09$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\"\"*&#F.\" \"$F.)F-F7F.F.*&#F.\"\"%F.*$)F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.*&#F.\"\"' F.*$)F-FDF.F.F4*&#F.\"\"(F.)F-FIF.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 142 "plot( [f(x), g(x), [[-1,0],[-1, g(-1)]], [[1,0],[1 ,g(1)]] ], x = -3..3, y = -7..3, color = [red, blue, coral, coral], th ickness = [3,1,2,2] );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7O7$$!3S+++&[Ib***!#=$!3EPgNq/18x!#<7$$!3s* *****\\XF`**F*$!3S1B'yQ^gO&F-7$$!3[*****pAyu\"**F*$!3w.NI)>ysz%F-7$$!3 E+++.>o\"))*F*$!3fZ`?PO'pV%F-7$$!3/+++!e&)e%)*F*$!3awxUe]ksTF-7$$!3y** ***pD*35)*F*$!3)H8yQ='yjRF-7$$!3L+++5m\\Q(*F*$!3*f5%)[7$*Qk$F-7$$!3w** ***H'R!pm*F*$!3M^aS$F-7$$!3$)******p'=P_*F*$!3X#4AHmJV/$F-7$$!3s* ****\\PL0Q*F*$!3-c-XY:[\"y#F-7$$!3y*****zyiT4*F*$!35q8yf2[,CF-7$$!3u** *****>#z2))F*$!3,P`o5#yn7#F-7$$!3'******\\0e:9)F*$!3E[RG?m%Go\"F-7$$!3 =+++5RKvuF*$!3'od$o>BZw8F-7$$!3%)*****\\!Qf&)oF*$!3wFZ`'eYl;\"F-7$$!3] *******pjeH'F*$!3-!*)4$)y\\8$**F*7$$!3h******4*3=+&F*$!3+9%)=!G!4NpF*7 $$!3!)******RFcpPF*$!3EIzhkdQJZF*7$$!3'*******>J%Q[#F*$!3!Rxf&)Q,`&GF* 7$$!34+++g6:.8F*$!3cVXV_LC'R\"F*7$$!35++++!Q:'H!#?$!3%4s+7.Kf'HFjq7$$ \"3++++!RIKH\"F*$\"34F$Gns$=;7F*7$$\"3/+++5:xWCF*$\"3-P.tE[:(=#F*7$$\" 3-+++![n%)o$F*$\"3,Z.bbfoRJF*7$$\"37++++rKt\\F*$\"3@PO7>L&o.%F*7$$\"3u ******z:JIiF*$\"3,W1Ih[&H%[F*7$$\"3%********o3lW(F*$\"3u+2l0YalbF*7$$ \"3!)******>#))oz)F*$\"3i'4\"HHC16jF*7$$\"3-+++Ik-,5F-$\"3H$o,\"REgOpF *7$$\"36+++D-eI6F-$\"3qT\")G([VRc(F*7$$\"3)*******=_(zC\"F-$\"3%*><%R3 *H+\")F*7$$\"3!******\\&*=jP\"F-$\"3u#Q3\"Gj_b')F*7$$\"31+++4/3(\\\"F- $\"3wj%=-8A7:*F*7$$\"35+++C4JB;F-$\"3(*p879BPW'*F*7$$\"3)******\\KCnu \"F-$\"3o*eWr/4/,\"F-7$$\"3'*******=n#f(=F-$\"3AoLNf\\Pc5F-7$$\"3$**** ***zRO+?F-$\"3i\"HvzgL()4\"F-7$$\"3,+++_!>w7#F-$\"3r![@)G?FS6F-7$$\"3# *******)Q?QD#F-$\"379^G5)H)z6F-7$$\"3%)******4jypBF-$\"3Oa#eGL\\[@\"F- 7$$\"38+++Ujp-DF-$\"3=V4LbI``7F-7$$\"3++++gEd@EF-$\"3gp:'zO3pG\"F-7$$ \"3;+++4'>$[FF-$\"32LpGNwI@8F-7$$\"35+++6EjpGF-$\"3Mqr(*p&fJN\"F-7$$\" \"$\"\"!$\"3c!*)>6O%H'Q\"F--%'COLOURG6&%$RGBG$\"*++++\"!\")$FcyFcyF]z- %*THICKNESSG6#Fby-F$6%7W7$$!\"$Fcy$!3a8dG9dy#>&!#:7$$!3&*****\\P&3Y$HF -$!3.7SLV'4&>XFiz7$$!3!******\\2<#pGF-$!3yk.KHlqBRFiz7$$!3')**\\78.K7G F-$!3%3%3?seNiMFiz7$$!3#)***\\7bBav#F-$!3Q(Qi%f<>\\IFiz7$$!3'***\\(=>P 9p#F-$!3\"pPsrfZnj#Fiz7$$!36++]K3XFEF-$!3q)f&Q?D7uAFiz7$$!3w******H./j DF-$!3?Eg8Sl:a>Fiz7$$!3%)****\\F)H')\\#F-$!3uYchifdu;Fiz7$$!3#****\\i3 @/P#F-$!3tVN$*G:?@7Fiz7$$!3;++Dr^b^AF-$!38QQc1R&4-*!#;7$$!3$****\\7Sw% G@F-$!3K3_cm_wClF\\^l7$$!3*****\\7;)=,?F-$!3BQhj(3q!f\"F\\^l7$$!3!******\\!)H%*\\\"F-$!3wo ,x#>Lb3\"F\\^l7$$!3/+++vl[p8F-$!3/xb!o9GjR(F-7$$!3\"******\\>iUC\"F-$! 3%[5j3@YU8&F-7$$!3-++DhkaI6F-$!3)\\I+U4!H9PF-7$F/$!3%*H&f\\Qx.c#F-7$F[ o$!3ysW2xWX*)=F-7$$!3S++]7RKvuF*$!3;WN\">yV'Q8F-7$$!3s,+++P'eH'F*$!3=P *3yYW+')*F*7$$!3q)***\\7*3=+&F*$!3ylT,A#4i#pF*7$$!3[)***\\PFcpPF*$!3iT YIuqhIZF*7$$!3;)****\\7VQ[#F*$!3QMsEK\"y_&GF*7$$!32)***\\i6:.8F*$!3C&* QqPKC'R\"F*7$$!3Wb+++v`hHFjq$!3eBcrH:$f'HFjq7$$\"3]****\\(QIKH\"F*$\"3 =M(QA\"Q=;7F*7$$\"38****\\7:xWCF*$\"3M`LUQz;(=#F*7$$\"3E,++vuY)o$F*$\" 3Ms-F/(3+9$F*7$$\"3!z******4FL(\\F*$\"3hx%pmZ,,/%F*7$$\"3A)****\\d6.B' F*$\"3k)fU(*f\\7'[F*7$$\"3s****\\(o3lW(F*$\"37omv=^!oj&F*7$$\"35***** \\A))oz)F*$\"3CotvSKGjlF*7$$\"3e******Hk-,5F-$\"3*z[%4J#=bg(F*7$Fft$\" 3-S<+q8$RB*F*7$$\"3u***\\(=_(zC\"F-$\"3e3fH#oN+;\"F-7$$\"3M+++b*=jP\"F -$\"3#40V\\4ZCf\"F-7$$\"3g***\\(3/3(\\\"F-$\"3m3$4DYPXF#F-7$$\"33++vB4 JB;F-$\"3&*)*[eT=wVMF-7$$\"3u*****\\KCnu\"F-$\"3m:E(G1kfF&F-7$$\"3s*** \\(=n#f(=F-$\"3M[SUCu$HG)F-7$$\"3P+++!)RO+?F-$\"3'\\#*3rVP,F\"F\\^l7$$ \"30++]_!>w7#F-$\"3hc2&R=Z0%>F\\^l7$$\"3O++v)Q?QD#F-$\"3eDunpyt1HF\\^l 7$$\"3G+++5jypBF-$\"3!*yM-=*o-:%F\\^l7$$\"3<++]Ujp-DF-$\"3E!**R)G1XKhF \\^l7$Fbx$\"3O!=f$*eM$[FF-$\"3(=n/M.dN?\"Fiz7$$\" 37++D6EjpGF-$\"3*\\9\\#f6ZW;Fiz7$Fay$\"3#Qr&G9dynAFiz-Fgy6&FiyF]zF]zFj y-F_z6#\"\"\"-F$6%7$7$$!\"\"FcyF]z7$F^[m$!3'H9dG9dGf#F--Fgy6&FiyFjy$\" )AR!)\\F\\zF]z-F_z6#\"\"#-F$6%7$7$$FijlFcyF]z7$F^\\m$\"3!\\4Q_4Q_f(F*F c[mFg[m-%+AXESLABELSG6$Q\"x6\"Q\"yFf\\m-%%VIEWG6$;FezFay;$!\"(FcyFay" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "A power series does not necessarily represent the function for all values of x. For example, " }{XPPEDIT 18 0 "xk = 1/ (1-x)" "6#/%#xkG*&\"\"\"F&,&F&F&%\"xG!\"\"F)" }{TEXT -1 311 " only for | x | < 1. Every power series has an interval of convergence - althou gh in some cases it is all real numbers or just a single number. In th e process we underwent to find his series, one of the steps included e xpanding a power series. Thus that interval of convergence effects the final series outcome." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "We can examine this concept in further detail using \+ a customized plot function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1028 "conv_plot := proc( \+ f, g1,g2, a,b, c1, c2, eps)\nlocal x1,x2,y1,y2,delta, m, M, A, B, i, n , CL;\nn := 1: delta := (b-a)/n: x2 := a: \nM := maximize( f(x), x = a..b): m := minimize(f(x), x = a..b): \n\nfor i fr om 1 to n do\nx1 := x2: x2:= x1 + delta: y1 := evalf( f(x1)): \+ y2 := evalf( f(x2)):\nif( abs(evalf( y1 - g1(x1))) < eps ) then \+ \n CL := maroon: \nelse \n CL := black: \nfi:\n#print ( [[x1, m ],[x1,y1],[x2,y2],[x2,m]]); \nA[i] := polygonplot( [[x1, m],[x1,y1],[x 2,y2],[x2,m]], color = CL, style = patchnogrid):\nif( abs(evalf( y1 - \+ g2(x1))) < eps ) then \n CL := coral: \nelse \n CL := black: \nf i: \nB[i] := polygonplot( [[x1, M],[x1,y1],[x2,y2],[x2,M]], color = \+ CL, style = patchnogrid):\nod:\n\ndisplay( [plot([f(x), g1(x), g2(x)], x = a..b, y = n..M, thickness = [4,2,2], color = [red, blue, green] ) ,\n plot( [[[c1, m],[c1,g1(c1)]], [[c2, g2(c2)],[c2,M]] ] \+ , x =a..b, color = [blue, green], linestyle = 3),\n seq( A[i], i = 1..n), seq( B[i], i = 1..n)], axes = framed ); \nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 292 "This special plot f uncion will show where the series is within a small tolerance of the \+ original function. This will provide us a convenient way to see where \+ the series converges to the function and where it doesn't. It allows u s to compare two approximate functions to an original function." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> ln( 1+x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#% \"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,&\"\"\"F09$F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "g := unapply( convert(series( f(x), x, 7), polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"x G6\"6$%)operatorG%&arrowGF(,.9$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\"\"*&#F. \"\"$F.)F-F7F.F.*&#F.\"\"%F.*$)F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.*&#F.\" \"'F.*$)F-FDF.F.F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h := unapply( convert(series( f(x), x, 12), polynom), x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(,89$ \"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\"\"*&#F.\"\"$F.)F-F7F.F.*&#F.\"\"%F.*$) F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.*&#F.\"\"'F.*$)F-FDF.F.F4*&#F.\"\"(F.)F -FIF.F.*&#F.\"\")F.*$)F-FMF.F.F4*&#F.\"\"*F.)F-FRF.F.*&#F.\"#5F.*$)F-F VF.F.F4*&#F.\"#6F.)F-FenF.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot( \{f(x), g(x)\}, x = -3..3, y = -6..3);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7fn7 $$!\"$\"\"!$!3%***********\\o?!#:7$$!3)****\\(oUInH!#<$!3!*3zkiEP_>F-7 $$!3&*****\\P&3Y$HF1$!3D\\ei=M'=%=F-7$$!3#****\\i!G\">!HF1$!3t+bK.#\\n t\"F-7$$!3!******\\2<#pGF1$!3W\"Gk2t8oj\"F-7$$!3')**\\78.K7GF1$!3uRH+! yMYZ\"F-7$$!3#)***\\7bBav#F1$!3jE2HX'[kK\"F-7$$!3'***\\(=>P9p#F1$!3aK' Rc`r_<\"F-7$$!36++]K3XFEF1$!3WH\"R9R(=R5F-7$$!3w******H./jDF1$!3KgZ_1- uh\"*!#;7$$!3%)****\\F)H')\\#F1$!3UAMb;\"[)f!)Ffn7$$!3))**\\(oXDXV#F1$ !3%z`j\"yqVzqFfn7$$!3#****\\i3@/P#F1$!3X$f#3f$*p/iFfn7$$!3;++Dr^b^AF1$ !3()y2RPiJI[Ffn7$$!3$****\\7Sw%G@F1$!3%>%*Rd^(Q(p$Ffn7$$!3*****\\7;)=, ?F1$!3Chr0hy#3y#Ffn7$$!3/++DO\"3V(=F1$!3_`&H#\\iKw?Ffn7$$!3#******\\V' zViUC\"F1$!3#y6n89RYZ%F17$$!3-++DhkaI6F1$!3Ed=9RU1xLF17$$!3s****** \\XF`**!#=$!3.Aiw'zF@U#F17$$!3u*******>#z2))F[s$!3cs$eLT42$=F17$$!3S++ ]7RKvuF[s$!3:lPOD&4+K\"F17$$!3s,+++P'eH'F[s$!3E?G&HUJS!)*F[s7$$!3q)*** \\7*3=+&F[s$!3J$fvF@?]\"pF[s7$$!3[)***\\PFcpPF[s$!3^D!Rw.s!HZF[s7$$!3; )****\\7VQ[#F[s$!3c%)p23[>bGF[s7$$!32)***\\i6:.8F[s$!3)eRvfKUiR\"F[s7$ $!3Wb+++v`hH!#?$!39BcrH:$f'HFdu7$$\"3]****\\(QIKH\"F[s$\"3t<4+[H=;7F[s 7$$\"38****\\7:xWCF[s$\"3=EeVnL4(=#F[s7$$\"3E,++vuY)o$F[s$\"3Z^Jl[=oQJ F[s7$$\"3!z******4FL(\\F[s$\"39c![c\"4NHSF[s7$$\"3A)****\\d6.B'F[s$\"3 n:&Hil$>4[F[s7$$\"3s****\\(o3lW(F[s$\"3++lw7#F1$!3CbP%)Q8o)y)F17$$\"3O++v)Q?QD#F1$!3G#)*Qu[,NJ\"Ff n7$$\"3G+++5jypBF1$!375t*)R^zX=Ffn7$$\"3<++]Ujp-DF1$!3+j+Yg%**Gl#Ffn7$ $\"3++++gEd@EF1$!3%otfJ!G^&f$Ffn7$$\"39++v3'>$[FF1$!3+(p*>F)[L)[Ffn7$$ \"39+++5h(*3GF1$!3+>$\\ny&*3i&Ffn7$$\"37++D6EjpGF1$!3CDY>@0TZkFfn7$$\" 31+]i0j\"[$HF1$!3?`b^+VRWuFfn7$$\"\"$F*$!3e++++++l&)Ffn-%'COLOURG6&%$R GBG$\"#5!\"\"$F*F*F`^l-F$6$7O7$$!3S+++&[Ib***F[s$!3EPgNq/18xF17$Fir$!3 S1B'yQ^gO&F17$$!3[*****pAyu\"**F[s$!3w.NI)>ysz%F17$$!3E+++.>o\"))*F[s$ !3fZ`?PO'pV%F17$$!3/+++!e&)e%)*F[s$!3awxUe]ksTF17$$!3y*****pD*35)*F[s$ !3)H8yQ='yjRF17$$!3L+++5m\\Q(*F[s$!3*f5%)[7$*Qk$F17$$!3w*****H'R!pm*F[ s$!3M^aS$F17$$!3$)******p'=P_*F[s$!3X#4AHmJV/$F17$$!3s*****\\PL0Q *F[s$!3-c-XY:[\"y#F17$$!3y*****zyiT4*F[s$!35q8yf2[,CF17$F_s$!3,P`o5#yn 7#F17$$!3'******\\0e:9)F[s$!3E[RG?m%Go\"F17$$!3=+++5RKvuF[s$!3'od$o>BZ w8F17$$!3%)*****\\!Qf&)oF[s$!3wFZ`'eYl;\"F17$$!3]*******pjeH'F[s$!3-!* )4$)y\\8$**F[s7$$!3h******4*3=+&F[s$!3+9%)=!G!4NpF[s7$$!3!)******RFcpP F[s$!3EIzhkdQJZF[s7$$!3'*******>J%Q[#F[s$!3!Rxf&)Q,`&GF[s7$$!34+++g6:. 8F[s$!3cVXV_LC'R\"F[s7$$!35++++!Q:'HFdu$!3%4s+7.Kf'HFdu7$$\"3++++!RIKH \"F[s$\"34F$Gns$=;7F[s7$$\"3/+++5:xWCF[s$\"3-P.tE[:(=#F[s7$$\"3-+++![n %)o$F[s$\"3,Z.bbfoRJF[s7$$\"37++++rKt\\F[s$\"3@PO7>L&o.%F[s7$$\"3u**** **z:JIiF[s$\"3,W1Ih[&H%[F[s7$$\"3%********o3lW(F[s$\"3u+2l0YalbF[s7$$ \"3!)******>#))oz)F[s$\"3i'4\"HHC16jF[s7$$\"3-+++Ik-,5F1$\"3H$o,\"REgO pF[s7$F`x$\"3qT\")G([VRc(F[s7$$\"3)*******=_(zC\"F1$\"3%*><%R3*H+\")F[ s7$$\"3!******\\&*=jP\"F1$\"3u#Q3\"Gj_b')F[s7$$\"31+++4/3(\\\"F1$\"3wj %=-8A7:*F[s7$$\"35+++C4JB;F1$\"3(*p879BPW'*F[s7$$\"3)******\\KCnu\"F1$ \"3o*eWr/4/,\"F17$$\"3'*******=n#f(=F1$\"3AoLNf\\Pc5F17$$\"3$*******zR O+?F1$\"3i\"HvzgL()4\"F17$$\"3,+++_!>w7#F1$\"3r![@)G?FS6F17$$\"3#***** **)Q?QD#F1$\"379^G5)H)z6F17$$\"3%)******4jypBF1$\"3Oa#eGL\\[@\"F17$$\" 38+++Ujp-DF1$\"3=V4LbI``7F17$F\\\\l$\"3gp:'zO3pG\"F17$$\"3;+++4'>$[FF1 $\"32LpGNwI@8F17$$\"35+++6EjpGF1$\"3Mqr(*p&fJN\"F17$Fe]l$\"3c!*)>6O%H' Q\"F1-Fj]l6&F\\^lF`^lF]^lF`^l-%+AXESLABELSG6$Q\"x6\"Q\"yFa\\m-%%VIEWG6 $;F(Fe]l;$!\"'F*Fe]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 13 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 174 " \+ original function centers of series ex pansions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 670 "The original function, f(x0 = ln(1+x) is the thick red curve. The value curve is g(x) which is the series taken out to 7 terms. The int erval where g differs from f by less than epsilon = .05 is indicated \+ by the purple shading on the bottom of the screen, The dashed blue lin e indicates the center of the expansion. In a similar way, the green c urve is the graph of h(x), which is the series taken out to 12 terms. \+ The interval where it is close to f(x) is indicated by the yellow bar \+ above the graph. What this diagram shows is that the series with more \+ terms is a better fit to the function.Also, both series seem to conver ge within the same region between - 1 and +1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "Lets look at another exa mple. We'll consider tan-1(x), and look at series expansions of 3 and \+ 15 terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := x -> arctan(x); c := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%'arctanG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" cG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "series( D(f)(x), x = c, 3); convert(int(%, x), polynom); g := unapply( %,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"xG\"\"\"\"\"!!\"\"\"\"#-%\"OG6#F% \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"*&#F%\"\"$F%*$)F $F(F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)o peratorG%&arrowGF(,&9$\"\"\"*&#F.\"\"$F.*$)F-F1F.F.!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "series( D(f)(x), x = c, 15); convert(int(%, x), polynom); h := unapply( %,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!!\"\"\"\"#F%\"\"%F'\"\"'F%\"\")F '\"#5F%\"#7F'\"#9-%\"OG6#F%\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2% \"xG\"\"\"*&#F%\"\"$F%*$)F$F(F%F%!\"\"*&#F%\"\"&F%)F$F.F%F%*&#F%\"\"(F %*$)F$F2F%F%F+*&#F%\"\"*F%)F$F7F%F%*&#F%\"#6F%*$)F$F;F%F%F+*&#F%\"#8F% )F$F@F%F%*&#F%\"#:F%*$)F$FDF%F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(,29$\"\"\"*&#F.\"\"$F.*$)F-F1F. F.!\"\"*&#F.\"\"&F.)F-F7F.F.*&#F.\"\"(F.*$)F-F;F.F.F4*&#F.\"\"*F.)F-F@ F.F.*&#F.\"#6F.*$)F-FDF.F.F4*&#F.\"#8F.)F-FIF.F.*&#F.\"#:F.*$)F-FMF.F. F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plot( [f(x),g(x ), h(x) ], x = -5..5, y = -4..4, color = [red, blue, green], thickness = [3,1,2] );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 '-%'CURVESG6%7S7$$!\"&\"\"!$!3!f,Xpw+MP\"!#<7$$!3YLLLe%G?y%F-$!3'zWwsn ]YO\"F-7$$!3OmmT&esBf%F-$!3]t4++4Rc8F-7$$!3ALL$3s%3zVF-$!3GA&=')y(GY8F -7$$!3_LL$e/$QkTF-$!3r1g*)3p7N8F-7$$!3ommT5=q]RF-$!37,?HYP)GK\"F-7$$!3 ILL3_>f_PF-$!3Re3)Q!fO58F-7$$!3K++vo1YZNF-$!3/];MSU.'H\"F-7$$!3;LL3-OJ NLF-$!3yIjn^H]z7F-7$$!3p***\\P*o%Q7$F-$!3S(fujB')4E\"F-7$$!3Kmmm\"RFj! HF-$!3*Q6Z>'zSR7F-7$$!33LL$e4OZr#F-$!3PBD>d(fy@\"F-7$$!3u*****\\n\\!* \\#F-$!3q$R/u#)e,>\"F-7$$!3%)*****\\ixCG#F-$!3spaOy_'y:\"F-7$$!3#***** *\\KqP2#F-$!3Gv]]:*y97\"F-7$$!39LL3-TC%)=F-$!3AC7'oIyG3\"F-7$$!3[mmm\" 4z)e;F-$!39uku0$3$G5F-7$$!3Mmmmm`'zY\"F-$!3v:daYK*ys*!#=7$$!3#****\\(= t)eC\"F-$!3'>L*y3@ZW*)F]q7$$!3!ommmh5$\\5F-$!31,y@Adb%4)F]q7$$!3S$*** \\(=[jL)F]q$!3#3yqjvh\"\\pF]q7$$!3)f***\\iXg#G'F]q$!3#*)=Y8y/%4cF]q7$$ !3ndmmT&Q(RTF]q$!3)HjwggW\\#RF]q7$$!3%\\mmTg=><#F]q$!3H-^F>BqQ@F]q7$$! 3vDMLLe*e$\\!#?$!3$G&zR\\d&e$\\F_s7$$\"3!=nm\"zRQb@F]q$\"3xN=5Jr!H7#F] q7$$\"3_,+](=>Y2%F]q$\"3Up=L$oC#pQF]q7$$\"3summ\"zXu9'F]q$\"3RQCBwh!>^ &F]q7$$\"3#4+++]y))G)F]q$\"3)ysp%)p*3@pF]q7$$\"3H++]i_QQ5F-$\"3gqf'p[r A/)F]q7$$\"3b++D\"y%3T7F-$\"3h<7%p#4hD*)F]q7$$\"3+++]P![hY\"F-$\"3xW\\ `u!G@s*F]q7$$\"3iKLL$Qx$o;F-$\"3nBtu@$H3.\"F-7$$\"3Y+++v.I%)=F-$\"3Wbb At1*G3\"F-7$$\"3?mm\"zpe*z?F-$\"3_EHP(fVE7\"F-7$$\"3;,++D\\'QH#F-$\"3Y N^\"RU\"pf6F-7$$\"3%HL$e9S8&\\#F-$\"3'pV98l<'*=\"F-7$$\"3s++D1#=bq#F-$ \"3ftLhE^v;7F-7$$\"3\"HLL$3s?6HF-$\"3[k*G&[P#*R7F-7$$\"3a***\\7`Wl7$F- $\"3!4%Gv'yO7E\"F-7$$\"3enmmm*RRL$F-$\"39'=YHh*Qz7F-7$$\"3%zmmTvJga$F- $\"3_I#pP,HfH\"F-7$$\"3]MLe9tOcPF-$\"3*e$oC#*fh58F-7$$\"31,++]Qk\\RF-$ \"3$\\ws&G+#GK\"F-7$$\"3![LL3dg6<%F-$\"32[s$z%e\\N8F-7$$\"3%ymmmw(GpVF -$\"3%*R$\\&)=,eM\"F-7$$\"3C++D\"oK0e%F-$\"3e)=C**e`eN\"F-7$$\"35,+v=5 s#y%F-$\"3A^?x*ozYO\"F-7$$\"\"&F*$\"3!f,Xpw+MP\"F--%'COLOURG6&%$RGBG$ \"*++++\"!\")$F*F*Fa[l-%*THICKNESSG6#\"\"$-F$6%7S7$F($\"3WmmmmmmmO!#;7 $F/$\"3U+gJdb%p;$F\\\\l7$F4$\"3;l.P7K=pFF\\\\l7$F9$\"3%\\&\\l'Qe7O#F\\ \\l7$F>$\"37+7]!ok3*>F\\\\l7$FC$\"3M\"e\"*)zTNg;F\\\\l7$FH$\"3%z9?tv+i Q\"F\\\\l7$FM$\"3(>&o\"G'=NL6F\\\\l7$FR$\"327(40Z\"QK!*F-7$FW$\"3!\\v^ $*\\Au.(F-7$Ffn$\"3[_\\UF)omF&F-7$F[o$\"33')oD@uFaRF-7$F`o$\"3Ug7f%QYL q#F-7$Feo$\"3_(>O>1+7o\"F-7$Fjo$\"3y/vLUe&**)*)F]q7$F_p$\"3oC^&y23oX$F ]q7$Fdp$!39.?B\"[/?P\"F]q7$Fip$!3$HiqBB\\^8%F]q7$F_q$!3Ln7Sqf]7gF]q7$F dq$!3Cq-*)G;&>k'F]q7$Fiq$!3i?tbN9C0kF]q7$F^r$!33J:Jl(**fX&F]q7$Fcr$!3k &4\\<(oD.RF]q7$Fhr$!3yv&zd8nx8#F]q7$F]s$!3b8?\")[d&e$\\F_s7$Fcs$\"3wAt Jkl+A@F]q7$Fhs$\"3?lIIyE7\\QF]q7$F]t$\"3s4V%f?]IP&F]q7$Fbt$\"3YyV=tJd! R'F]q7$Fgt$\"3I;YWcQu^mF]q7$F\\u$\"3ZSrq'GS(QgF]q7$Fau$\"3sLOB5&*3cTF] q7$Ffu$\"3fYp%zO1T?\"F]q7$F[v$!3W**>R3LCeMF]q7$F`v$!3c5]J'\\**\\>*F]q7 $Fev$!3`mx(zoL%Hd-mg%Go#F-7$F_w$!3K*)R%**y\"z&*QF-7$Fdw$! 3:4[.%zwIJ&F-7$Fiw$!3O(eg&zA2hqF-7$F^x$!3X5Q!Gk![=!*F-7$Fcx$!3c.%)>ksp J6F\\\\l7$Fhx$!3;V+z;]9\"R\"F\\\\l7$F]y$!37x0t[\"4)e;F\\\\l7$Fby$!3ZLF l4(f>+#F\\\\l7$Fgy$!3sxLzmI\\VBF\\\\l7$F\\z$!3]T6+-8YXFF\\\\l7$Faz$!3q *R&pj-YoJF\\\\l7$Ffz$!3WmmmmmmmOF\\\\l-F[[l6&F][lFa[lFa[lF^[l-Fc[l6#\" \"\"-F$6%7\\o7$F($\"3&oK&*oWP[%>F`[l7$$!39LLe9r]X\\F-$\"3G-M&*>kV[;F`[ l7$$!3Gmm;HU,\"*[F-$\"3k%zgUX@YR\"F`[l7$$!3J++vV8_O[F-$\"3kd*[F<`w<\"F `[l7$F/$\"3e\"eTBHN^#**!\"*7$$!3!***\\(=_+so%F-$\"3&y:h^1>^L(Fjfl7$F4$ \"3QTk)fgdrQ&Fjfl7$$!3O**\\7`'Gd[%F-$\"3+^Y3dphxPFjfl7$F9$\"3=yJPC48EE Fjfl7$$!3#HLLL)QtrUF-$\"3d2DV#H%f/=Fjfl7$F>$\"3T1%\\0\")*4G7Fjfl7$$!35 +]7GCadSF-$\"3y:rD$*oe)G)!#57$FC$\"3wP,$G$zsMbFhhl7$FH$\"3w8S.+8URDFhh l7$FM$\"33>!ov:UI3\"Fhhl7$FR$\"3oe'G:,J![U!#67$FW$\"3s?1ofBvp:Feil7$Ff n$\"3!Q)z\")QrJJ_!#77$F[o$\"3wz6![p_&[=F\\jl7$F`o$\"3`jAs_I`8_!#87$Feo $\"3?8#)[H\"y$)H\"Fcjl7$Fjo$\"3cSX!>O*HnH!#97$F_p$\"3,Ve;6Q[Rn!#:7$Fdp $\"3(zFG;q_pA*F\\\\l7$Fip$\"3CS\"f_L)f\"G\"F\\\\l7$F_q$\"3y;#p%RfxQ9F] q7$Fdq$!3'pmQbG.4U(F]q7$Fiq$!3c/>j(f$oKpF]q7$F^r$!3A#)z^XPC4cF]q7$Fcr$ !32*pf*[W%\\#RF]q7$Fhr$!3k'4s#>BqQ@F]q7$F]s$!3'>&zR\\d&e$\\F_s7$Fcs$\" 35&>*4Jr!H7#F]q7$Fhs$\"3oF#QGcC#pQF]q7$F]t$\"3,q1VnOz6bF]q7$Fbt$\"37$Fau$!33xW:!)=6b 7F\\\\l7$Ffu$!3S)3`()4%[45F^[m7$F[v$!37nMjgkgUnF^[m7$F`v$!3j_[@90'o5$F jjl7$Fev$!3UP8*\\*)>:S\"Fcjl7$Fjv$!35)zg!=K&)*3&Fcjl7$F_w$!3)\\P$*>r#* \\v\"F\\jl7$Fdw$!3G'o_%f3ym`F\\jl7$Fiw$!3AvPR&H)\\!f\"Feil7$F^x$!3uG*G xzP:A%Feil7$Fcx$!3/%*RbEUWw5Fhhl7$Fhx$!3)[;pu(eSyDFhhl7$$\"3yn;H#e0I&Q F-$!3Y.[7)eT$)y$Fhhl7$F]y$!3:j%zqjMB^&Fhhl7$$\"3%zm;/@-/1%F-$!35yFk.(= uP)Fhhl7$Fby$!3#*e4/6!z'e7Fjfl7$$\"3K,+voTAqUF-$!3Q#Rv)Qf(\\z\"Fjfl7$F gy$!3trI]F`[l-F[[l6 &F][lFa[lF^[lFa[l-Fc[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yFddm-%%VIEWG6$ ;F(Ffz;$!\"%F*$\"\"%F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "At first glance, the blue grap h ( the series with 3 terms ) might seem to mimize the function. Howev er, on closer inspection, we see that within the interval [-1, 1], the green curve ( the series with 15 terms) stays with the curve better. \+ We can see this even better using the special plot command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "co nv_plot( f, g, h, -3, 3, c, c, .1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$!\"$\"\"!$!3Wa#)Rsd/\\7!#<7$$! 3!******\\2<#pGF-$!3Q^D#3CMaB\"F-7$$!3#)***\\7bBav#F-$!3:Qn\"F-7$ $!3#****\\i3@/P#F-$!3G'\\X'>Ker6F-7$$!3;++Dr^b^AF-$!3YA!HBPGG:\"F-7$$! 3$****\\7Sw%G@F-$!3Z@gB$=$eJ6F-7$$!3*****\\7;)=,?F-$!3a,^:PiQ26F-7$$!3 /++DO\"3V(=F-$!3e$Q75u&o!3\"F-7$$!3#******\\V'zV,0\"F-7$$ !3******\\d;%)G;F-$!3%=9%ycZ>?5F-7$$!3!******\\!)H%*\\\"F-$!3'Qs.XK#=E )*!#=7$$!3/+++vl[p8F-$!3!*z$zJ!o(3S*Fdo7$$!3\"******\\>iUC\"F-$!3b1uN, '*4Q*)Fdo7$$!3-++DhkaI6F-$!3S:e;2N&fY)Fdo7$$!3s******\\XF`**Fdo$!35&Hr ACk0$yFdo7$$!3u*******>#z2))Fdo$!39qz@o#Q4A(Fdo7$$!3S++]7RKvuFdo$!3*[F Lsn*>>kFdo7$$!3s,+++P'eH'Fdo$!3([[$)*He!*=cFdo7$$!3q)***\\7*3=+&Fdo$!3 muiH:J#zj%Fdo7$$!3[)***\\PFcpPFdo$!3*=d>abY[g$Fdo7$$!3;)****\\7VQ[#Fdo $!3&\\`(H*QuXV#Fdo7$$!32)***\\i6:.8Fdo$!3IXXX3([eH\"Fdo7$$!3Wb+++v`hH! #?$!3s$**)yT)G:'HF_s7$$\"3]****\\(QIKH\"Fdo$\"3%*>!3zN#4'G\"Fdo7$$\"38 ****\\7:xWCFdo$\"3I1i.8(RxR#Fdo7$$\"3E,++vuY)o$Fdo$\"3'Q2v>b^O`$Fdo7$$ \"3!z******4FL(\\Fdo$\"3NFu)>,:^h%Fdo7$$\"3A)****\\d6.B'Fdo$\"31]6GbE# =d&Fdo7$$\"3s****\\(o3lW(Fdo$\"3(H]sml)o+kFdo7$$\"35*****\\A))oz)Fdo$ \"3H@M]KXz9sFdo7$$\"3e******Hk-,5F-$\"3!))Gz4:6\"fyFdo7$$\"36+++D-eI6F -$\"3Y%=/x'=5F-7$$\"3u*****\\KCnu\"F-$\"39N-N`F%30\"F-7 $$\"3s***\\(=n#f(=F-$\"3'))G7l9W53\"F-7$$\"3P+++!)RO+?F-$\"3k*)p>.:A26 F-7$$\"30++]_!>w7#F-$\"3u!=rw5G98\"F-7$$\"3O++v)Q?QD#F-$\"3kc]6s7?`6F- 7$$\"3G+++5jypBF-$\"39Og>%H([r6F-7$$\"3<++]Ujp-DF-$\"3S[c]8:m!>\"F-7$$ \"3++++gEd@EF-$\"3\\h5a=DQ17F-7$$\"39++v3'>$[FF-$\"3cy&pr'*G=A\"F-7$$ \"37++D6EjpGF-$\"3yAUXV#zaB\"F-7$$\"\"$F*$\"3Wa#)Rsd/\\7F--%'COLOURG6& %$RGBG$\"*++++\"!\")$F*F*Fa[l-%*THICKNESSG6#\"\"%-F$6%7S7$F($\"\"'F*7$ F/$\"3>gD!eZ.V+&F-7$F4$\"3'y)*>fmkz@%F-7$F9$\"30/ZP#e\\(=MF-7$F>$\"3R! f#G?Y9,FF-7$FC$\"3S:eOLbHp?F-7$FH$\"3a?:@E$)>`:F-7$FM$\"3i*R%)o7:e3\"F -7$FR$\"3n%)4IW(RBq'Fdo7$FW$\"3b$z*fE0E0KFdo7$Ffn$\"3iH!zB$[2tB!#>7$F[ o$!3'G7&3ytM$)=Fdo7$F`o$!39vGGhD7dPFdo7$Ffo$!3d7y@Y>KL^Fdo7$F[p$!3n)4b geZ9-'Fdo7$F`p$!3?#H$=2o#))['Fdo7$Fep$!3-7h%*z'[km'Fdo7$Fjp$!3[`?>U#z, `'Fdo7$F_q$!3-umg$f3H3'Fdo7$Fdq$!31*HS)>V,kaFdo7$Fiq$!3U!Q!3a)*o%e%Fdo 7$F^r$!3)**G285;5f$Fdo7$Fcr$!3e/yH%3jFV#Fdo7$Fhr$!3u'QG`WudH\"Fdo7$F]s $!3MIMuT)G:'HF_s7$Fcs$\"35SD*e'3-'G\"Fdo7$Fhs$\"3+hM^mU1'R#Fdo7$F]t$\" 3dqR[G!)>@NFdo7$Fbt$\"3QCyS\\JHjXFdo7$Fgt$\"3&e.#*43wTU&Fdo7$F\\u$\"3! GuG>YL,2'Fdo7$Fau$\"3Al/@)zBx_'Fdo7$Ffu$\"3!z`wu7cmm'Fdo7$F[v$\"3_ItQ& )Gt)['Fdo7$F`v$\"3abfz3:#4+'Fdo7$Fev$\"3;K!ofakG2&Fdo7$Fjv$\"3kXWQ8rO' y$Fdo7$F_w$\"3C$*4sX'zU(>Fdo7$Fdw$!3O&*=:^>5sHFj]l7$Fiw$!3!>(3&Q7&)fC$ Fdo7$F^x$!3w>iX;()exmFdo7$Fcx$!3M[&oq*)*y#3\"F-7$Fhx$!3O\"\\mu=GCc\"F- 7$F]y$!3'\\/eG_kj1#F-7$Fby$!31+*\\OJ2Ds#F-7$Fgy$!3-\"pt=_QTQ$F-7$F\\z$ !3'\\YAF'=FrTF-7$Faz$!3uf?Q)H4t+&F-7$Ffz$!\"'F*-F[[l6&F][lFa[lFa[lF^[l -Fc[l6#\"\"#-F$6%7\\o7$F($\"3I/ZH0nS\"[)!#77$$!3)****\\(oUInHF-$\"3#yc IV`Ey<(Fhel7$$!3&*****\\P&3Y$HF-$\"3)R$**)R*>3jgFhel7$$!3#****\\i!G\"> !HF-$\"3U:i#30A:6&Fhel7$F/$\"3l!)HUjQu+VFhel7$$!3')**\\78.K7GF-$\"3?LB U%)>#)oJFhel7$F4$\"3)=)>'f6f)>BFhel7$$!3'***\\(=>P9p#F-$\"3dRn(fDg0i\" Fhel7$F9$\"3$y4^)pb-A6Fhel7$$!3w******H./jDF-$\"3#RDY-?Unn(!#87$F>$\"3 EI3*4*)Q,?&F`hl7$$!3))**\\(oXDXV#F-$\"3K![\\$>\"\\B\\$F`hl7$FC$\"3w3yl ;%R(>BF`hl7$FH$\"3wuG$3aqG0\"F`hl7$FM$\"3\"o]@4gQHV%!#97$FR$\"3R%o&=<[ '=r\"Fbil7$FW$\"3D)=M#fbu3i!#:7$Ffn$\"3s3b'zr:)=?Fiil7$F[o$\"3k`tibfa9 p!#;7$F`o$\"3^,J3')ykC=F`jl7$Ffo$\"3oH$*=Dna(o$F-7$F[p$\"3Ef=f8rgK7Fdo 7$F`p$!3Uf*H'H.f[iFdo7$Fep$!3$*ok/aO#=a(Fdo7$Fjp$!3%fZ!eg#Q2=(Fdo7$F_q $!3)Q^s%f(3kT'Fdo7$Fdq$!3f%o#*45R(=cFdo7$Fiq$!3'fpBUU>zj%Fdo7$F^r$!3*z )orAl%[g$Fdo7$Fcr$!3\"=Zo#*QuXV#Fdo7$Fhr$!3uWXX3([eH\"Fdo7$F]s$!3G$**) yT)G:'HF_s7$Fcs$\"37>!3zN#4'G\"Fdo7$Fhs$\"35uR,8(RxR#Fdo7$F]t$\"3*o,r# H:lLNFdo7$Fbt$\"3I\\@Xb;6:YFdo7$Fgt$\"3eZ&)fXBorbFdo7$F\\u$\"3w)>^9To! )R'Fdo7$Fau$\"3G1BvNDRvrFdo7$Ffu$\"3fs7F7sjUvFdo7$F[v$\"3u'*zWLFoZiFdo 7$F`v$!33FIT^Jv4hz:F`hl7$F]y$!38;XZW(G-J#F`hl7$$\"3B++DE8COCF-$!3RC'RT yf-`$F`hl7$Fby$!3(*eM558BJ`F`hl7$$\"33++D,X8iDF-$!3AYMa!=V`j(F`hl7$Fgy $!3+Ju'pmiU3\"Fhel7$$\"31+]PMh%\\o#F-$!3sG/33-)=c\"Fhel7$F\\z$!3%R!)[? zH-B#Fhel7$$\"39+++5h(*3GF-$!3LLKr$oG=6$Fhel7$Faz$!3f*e2,%*[-J%Fhel7$$ \"3))*\\P%eWA-HF-$!3%3#4p[W*)>^Fhel7$$\"31+]i0j\"[$HF-$!3G(Hwl=D'pgFhe l7$$\"3E+D\"G:3u'HF-$!3?Y5%e%[l\"=(Fhel7$Ffz$!3I/ZH0nS\"[)Fhel-F[[l6&F ][lFa[lF^[lFa[lF_el-F$6%7$7$Fa[lF+7$Fa[lFa[lF]el-%*LINESTYLEG6#Fgz-F$6 %7$F^dm7$Fa[lFhzFhcmF_dm-%)POLYGONSG6%7&7$F($!+sd/\\7!\"*Fjdm7$Ffz$\"+ sd/\\7F]em7$FfzF[em-F[[l6&F][lF*F*F*-%&STYLEG6#%,PATCHNOGRIDG-Fgdm6%7& 7$F(F_emFjdmF^emF^emFbemFdem-%*AXESSTYLEG6#%&FRAMEG-%+AXESLABELSG6%Q\" x6\"Q\"yFdfm%(DEFAULTG-%%VIEWG6$;F(Ffz;$\"\"\"F*F_em" 1 2 0 1 10 0 2 9 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 257 "In this example we have been u sing c = 0 as the center for our series expansion. We cal also do the \+ same thing for a different center value. Here is an example with c = P i/3. Notice how the entire convergence region moves along with the cen ter to the right." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := x -> ar ctan(x); c := Pi/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%'arcta nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,$%#PiG#\"\"\"\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "series( f(x) , x = c, 3): \+ convert( %, polynom): g := unapply( %, x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "series( f(x) , x = c, 15): convert( %, polynom ): h := unapply( %, x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "conv_plot( f, g, h, -4, 4, c, c, .1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6%7S7$$!\"%\"\"!$!3bK!o Om\"F-7$$!3BLLL8>1DBF-$!37%*>i*=F-$!38w:a>m*p5\"F-7$$!30++++@)f#=F -$!3]E!*H[(e(p5F-7$$!3-+++gi,f;F-$!3:sVUj[MG5F-7$$!3qmmm\"G&R2:F-$!3;j t1/Xh])*!#=7$$!3XLLLtK5F8F-$!3Yk#43tc/D*Fcp7$$!3eLLL$HsV<\"F-$!3+.8;RA @a')Fcp7$$!3+-++]&)4n**Fcp$!3J!Gfm\"Q]PyFcp7$$!37PLLL\\[%R)Fcp$!3iDi]j QO$)pFcp7$$!3G)*****\\&y!pmFcp$!3+O5*\\h&p\")eFcp7$$!3Y******\\O3E]Fcp $!3IW)*3J7KdYFcp7$$!3NKLLL3z6LFcp$!3/dG1!y/\")>$Fcp7$$!3sLLL$)[`PREg&3^JFcp7$$\"36HLLLm& z\"\\Fcp$\"3#R>QOCE1d%Fcp7$$\"3>(******z-6j'Fcp$\"3iBI?nWObeFcp7$$\"3q \"******4#32$)Fcp$\"3LvG'e@q=$pFcp7$$\"3r$*****\\#y'G**Fcp$\"39?h%oG$> =yFcp7$$\"3G******H%=H<\"F-$\"3=nq=+q4[')Fcp7$$\"35mmm1>qM8F-$\"3'z86# 4e(yF*Fcp7$$\"3%)*******HSu]\"F-$\"3o^Rpx?v])*Fcp7$$\"3'HLL$ep'Rm\"F-$ \"3ZV!)zX8mH5F-7$$\"3')******R>4N=F-$\"3k>ieaD&=2\"F-7$$\"3#emm;@2h*>F -$\"3(ou!4Y*oj5\"F-7$$\"3]*****\\c9W;#F-$\"32N[.@F*z8\"F-7$$\"3Lmmmmd' *GBF-$\"3'QsD8uD_;\"F-7$$\"3j*****\\iN7]#F-$\"3!)\\%H\"3.Y!>\"F-7$$\"3 aLLLt>:nEF-$\"3$G73$za377F-7$$\"35LLL.a#o$GF-$\"3\"G$fv-,*=B\"F-7$$\"3 ammm^Q40IF-$\"3__!*G!Qa&\\7F-7$$\"3y******z]rfJF-$\"3)Ry[S#eGk7F-7$$\" 3gmmmc%GpL$F-$\"39%G@R3O'z7F-7$$\"3/LLL8-V&\\$F-$\"3#zx>9O^@H\"F-7$$\" 3=+++XhUkOF-$\"3q^jLw!*Q/8F-7$$\"3=+++:oaUK>srF-7$F4$!3%4E!3F!eIv'F- 7$F9$!3!HU@x()*z%H'F-7$F>$!3QuWfYS^ZeF-7$FC$!3#>1PjE5jT&F-7$FH$!3&)yBc yr'*G]F-7$FM$!3k)QK\"*G20k%F-7$FR$!3))Gni=$eAD%F-7$FW$!3'G6pqC7*yQF-7$ Ffn$!3)y\\@')z1\"4NF-7$F[o$!3/yW\\@CL&>$F-7$F`o$!3-7$f7!**[bGF-7$Feo$! 3n&HD9\"\\_GDF-7$Fjo$!3t$['R=h'pA#F-7$F_p$!3[WXt*=HY'>F-7$Fep$!3a*GC] \"4%pm\"F-7$Fjp$!3T(G2#Rn(oU\"F-7$F_q$!3/l^h0&3;;\"F-7$Fdq$!3[Nqe'Q'[$ R*Fcp7$Fiq$!3k\\ly[P]!4(Fcp7$F^r$!3hyABS*f$H]Fcp7$Fcr$!30%p$yI-)e,$Fcp 7$Fhr$!3'GNju,2-H\"Fcp7$F]s$\"3Mwk!4uSzQ%!#>7$Fcs$\"3w%H_[R!G*3#Fcp7$F hs$\"3^8wE.*f`S$Fcp7$F]t$\"3DJ/k=**f+ZFcp7$Fbt$\"3H?W$zU<6!fFcp7$Fgt$ \"3wO$o`4v-%pFcp7$F\\u$\"3^*4f!fkK=yFcp7$Fau$\"3A.I]Y)pkk)Fcp7$Ffu$\"3 I)>WIZZ)e#*Fcp7$F[v$\"3%)>zM/L.v(*Fcp7$F`v$\"3*Q$*4!)f**>,\"F-7$Fev$\" 3Ly9,!)fNO5F-7$Fjv$\"3u>y7EK`Y5F-7$F_w$\"3k\\t;ir'R/\"F-7$Fdw$\"3@\"\\ ,!>-TG5F-7$Fiw$\"3-X\"*HNs+$)**Fcp7$F^x$\"3o)f7OEZ$f&*Fcp7$Fcx$\"3#[F) *=iY/**)Fcp7$Fhx$\"3;'QfG+!z!H)Fcp7$F]y$\"3E%4]?yU*GvFcp7$Fby$\"3[-*>v =!p:lFcp7$Fgy$\"31eEVJ=m#[&Fcp7$F\\z$\"3MS]'*\\/R\\UFcp7$Faz$\"3'=V')H SX:%HFcp7$Ffz$\"3+>w\\*=HrR\"Fcp-F[[l6&F][lFa[lFa[lF^[l-Fc[l6#\"\"#-F$ 6%7gn7$F($\"3QaW&eFjj(G!#67$$!3iLL$e%G?yRF-$\"3O\\hY*[Mpq#Fgel7$$!3ymm m\"p0k&RF-$\"3KB^07mzYDFgel7$$!3&*****\\P&3Y$RF-$\"3](R_g2yaR#Fgel7$$! 3cLLL$Q6G\"RF-$\"3'4(Q*oJHDD#Fgel7$$!3!******\\2<#pQF-$\"37XK70t/!*>Fg el7$F/$\"3'fgOpVfhv\"Fgel7$$!3=LL3#f\"p(y$F-$\"3!zGar(ylt:Fgel7$$!37++ ]g7Fgel7$F4$ \"3512!y*G$\"3Uh4rSw'R!RF[il7$FC$\"3 j%3\"z%))RiA#F[il7$FH$\"3')etl(>;SH\"F[il7$FM$\"3(4y9#eE+-s!#87$FR$\"3 $)yu9Mtl=QF`jl7$FW$\"3m\\]+\"p\"=l>F`jl7$Ffn$\"3)y3Y^'*Q4c*!#97$F[o$\" 3sV\\$)4$))R*[Fjjl7$F`o$\"3j>3p2P7-AFjjl7$Feo$\"39S&>-Y#**[$*!#:7$Fjo$ \"3Gr8W%f_4&QFd[m7$F_p$\"3(ou[Y4Ujh\"Fd[m7$Fep$\"3S*[aqT-m@&!#;7$Fjp$ \"3/+2iA/'=y\"F^\\m7$F_q$\"3%\\Y!Rsrb5SF-7$Fdq$\"3%*\\7!Q^;2,&Fcp7$Fiq $!3I\\00SYU\")RFcp7$F^r$!3'f75n@RK^%Fcp7$Fcr$!3EadPiEpNKFcp7$Fhr$!3'e) Q5n\"\\pt\"Fcp7$F]s$!3^5N]W%4JC%F_s7$Fcs$\"3\\\"Qo(fFE2=yFcp7$Fau$\"31mq=+q4[')Fcp7$Ffu$\"3U*QF#4e(yF *Fcp7$F[v$\"3?4O[&4_2&)*Fcp7$F`v$\"3B!)3j\"[h'H5F-7$Fev$\"3c;C#\\ed=2 \"F-7$Fjv$\"3qAU1]iW16F-7$F_w$\"3%Q)**>Gr$)Q6F-7$Fdw$\"3s$RO`U*\\r6F-7 $Fiw$\"3M%*Qk$*fxH7F-7$F^x$\"3&\\cfDki2S\"F-7$Fcx$\"3')*Gmop\">I?F-7$F hx$\"3o\\$*pLiVuTF-7$F]y$\"3=s>c)\\w=+\"F^\\m7$Fby$\"3?Y>%ee!=?HF^\\m7 $Fgy$\"3p0(4\\hUwV(F^\\m7$F\\z$\"3&\\\"4h?#*Q=>Fd[m7$Faz$\"3j5)=yZWY^% Fd[m7$Ffz$\"3/*R\"HG[[u5Fjjl-F[[l6&F][lFa[lF^[lFa[lF^el-F$6%7$7$$\"3k( f'>^v>Z5F-F+7$Febm$\"31?-IEz[%3)FcpF\\el-%*LINESTYLEG6#\"\"$-F$6%7$Fgb m7$FebmFhzF_bmFjbm-%)POLYGONSG6%7&7$F($!+kw\"eK\"!\"*Ffcm7$Ffz$\"+kw\" eK\"Ficm7$FfzFgcm-F[[l6&F][lF*F*F*-%&STYLEG6#%,PATCHNOGRIDG-Fccm6%7&7$ F(F[dmFfcmFjcmFjcmF^dmF`dm-%*AXESSTYLEG6#%&FRAMEG-%+AXESLABELSG6%Q\"x6 \"Q\"yF`em%(DEFAULTG-%%VIEWG6$;F(Ffz;$\"\"\"F*F[dm" 1 2 0 1 10 0 2 9 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 313 "There is another interesting aspect to this investigation. The function we studied above, f(x) = ln( 1 + \+ x ), has a vertical asymptote at x = -1. When we expanded a series abo ut x = 0, ouir interval of convergence was (-1,1). But what happens wh en we expand around other values of x, farther from the singularity?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f := x -> ln(1+x); c1 := 0; c2 := 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,&\"\" \"F09$F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "g := unapply(convert(series( f(x), x = c1,12), pol ynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)ope ratorG%&arrowGF(,89$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\"\"*&#F.\"\"$F.)F-F 7F.F.*&#F.\"\"%F.*$)F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.*&#F.\"\"'F.*$)F-FD F.F.F4*&#F.\"\"(F.)F-FIF.F.*&#F.\"\")F.*$)F-FMF.F.F4*&#F.\"\"*F.)F-FRF .F.*&#F.\"#5F.*$)F-FVF.F.F4*&#F.\"#6F.)F-FenF.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "h := unapply(convert(series( f(x), \+ x = c2, 12), polynom), x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGR6 #%\"xG6\"6$%)operatorG%&arrowGF(,<-%#lnG6#\"\"$\"\"\"*&#F1F0F19$F1F1# \"\"#F0!\"\"*&#F1\"#=F1*$),&F4F1F6F7F6F1F1F7*&#F1\"#\")F1)F=F0F1F1*&#F 1\"$C$F1*$)F=\"\"%F1F1F7*&#F1\"%:7F1)F=\"\"&F1F1*&#F1\"%uVF1*$)F=\"\"' F1F1F7*&#F1\"&4`\"F1)F=\"\"(F1F1*&#F1\"&)[_F1*$)F=\"\")F1F1F7*&#F1\"'Z rF1)F=\"#6F1F 1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "plot( [ f(x), g (x), h(x) ], x = -0.85..7, y = -2..3, color = [red, blue, green], thic kness = [3,1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$!3w*************\\)!#=$!31\")e)[)*>r*=!#<7$$!3Cn \"z%\\3Bs!)F*$!3'4c\"=R;AY;F-7$$!3gL$e*)phWk(F*$!3o2'3P\"e\"eW\"F-7$$! 3(**\\P%[Dp;sF*$!3+YCoT^%*y7F-7$$!3Kmm\"zRB*)y'F*$!3d!oD?#)yf8\"F-7$$! 3o\\(=n4CX/'F*$!3s%[f[pT[F*F*7$$!3/L3_&zC,I&F*$!3i[-po8\\]vF*7$$!3_m;a e]\"ei$F*$!3ZF*$!3e\"Qd$G-Ad@F*7$$!32Q$3x6#4I E!#>$!3k&QHcq(HlEFY7$$\"3oKe9wJ:#H\"F*$\"3!=z)>B)H_@\"F*7$$\"3t*\\7.vL C!HF*$\"35B\"RRj3$[DF*7$$\"3OLekB#)ynXF*$\"3u+uLArFiPF*7$$\"3N*\\iS)=! yA'F*$\"3dH](*R%39%[F*7$$\"3SomTv*H`$zF*$\"3\"pLsA;u=%eF*7$$\"3/L$3xk@ $R%*F*$\"3w]?k2\"Grk'F*7$$\"3;+]70gC86F-$\"3o-Z,%o**[*)F*7$$\"3=$ek)z $oef\"F-$\"3O5F-7$$\"3gmm;( =ZE#>F-$\"3?I/aw(*[s5F-7$$\"3+]7ya%yp4#F-$\"3a7R;WpUI6F-7$$\"3Mmm\"f; \"H^AF-$\"3iJmE*>_!z6F-7$$\"3+]7Gnmf?CF-$\"33WZ(***\\\")H7F-7$$\"31]P% =a:=e#F-$\"3!oack!)peF\"F-7$$\"3'p;zW_I+v#F-$\"3D-.\")zRw@8F-7$$\"3z;H d*Q/X!HF-$\"3Q%Ga$e38i8F-7$$\"3UL3x@`7rIF-$\"3$fEtIX>RS\"F-7$$\"33$ekB k(>WKF-$\"3EC))\\*y_bW\"F-7$$\"3U\\(=igd[R$F-$\"3wx@dJ^V![\"F-7$$\"3'Q $ek%\\uvb$F-$\"3*feVHd!z;:F-7$$\"36++DipnDPF-$\"3k`q6B3,`:F-7$$\"3Q*\\ i5VK,*QF-$\"3]Sp8&Q>se\"F-7$$\"33]7G`:D\\SF-$\"3\"fWK]-S#>;F-7$$\"3_+v V4i#fA%F-$\"3Y7a5Z?j`;F-7$$\"3Qmm\"fCwYQ%F-$\"3=TqL(=dNo\"F-7$$\"3I+]P %zvTb%F-$\"3g;PBS+b9IZA3x\"F-7$$\"3];zW,-oL]F-$\"3mkP!*RrN(z\"F-7$$\"35]i!> zJ))>&F-$\"3+htj`3OC=F-7$$\"3im;aewHg`F-$\"3/iV\"zF-7$$\"3wL3-F\\j eeF-$\"3&>T1F-7$$\"3UM[P-'F-$\"33!f!QFqH\\>F-7$$\"3)4+]AUqa< 'F-$\"3'*4JdB$o1(>F-7$$\"3!p;/\"[5O\\jF-$\"3Q%>:sP8Y*>F-7$$\"3WLL$o*3* []'F-$\"3zlCeF\\b:?F-7$$\"3%3D\"ya\"=2n'F-$\"3b_$\\EC5u.#F-7$$\"3g](=( **fVHoF-$\"3!o6@nZ!*y0#F-7$$\"\"(\"\"!$\"3uN)z;aT%z?F--%'COLOURG6&%$RG BG$\"*++++\"!\")$F[\\lF[\\lFe\\l-%*THICKNESSG6#\"\"$-F$6%7Z7$F($!3vmgI KHbQ=F-7$F>$!3#Gk(Q.\"3Q8\"F-7$FH$!3QCtCRco\\vF*7$FM$!3],-\")[@G.XF*7$ FR$!3kQUYD-Ad@F*FV7$Fgn$\"3^A\\@B)H_@\"F*7$F\\o$\"3iJ$**)o)3$[DF*7$Fao $\"3g-r>5bKiPF*7$Ffo$\"3glDAe7@V[F*7$F[p$\"395m1oX*=(eF*7$F`p$\"3!)HWz ruDqoF*7$Fep$\"330X-.w!Q(*)F*7$Fjp$\"3Ajd[W'Rxe\"F-7$F_q$\"3SfLzvpk4RF -7$Fdq$\"3*z'eqV;)p,\"!#;7$Fiq$\"39.Q2JC:bJFj_l7$F^r$\"3S(4-'\\Nb!z(Fj _l7$Fcr$\"3k`-n]%Q%p?!#:7$Fhr$\"3z@RP`P[9YFd`l7$F]s$\"3W%))pU%40Z5!#97 $Fbs$\"3'Gk!['>S!p@F[al7$Fgs$\"3N]q(ef\"RAWF[al7$F\\t$\"3IsOQ$=$R!>)F[ al7$Fat$\"3mG'HE[,b`\"!#87$Fft$\"3M'o`)H8CYGFhal7$F[u$\"3<+:6-,aUZFhal 7$F`u$\"3/l1gQQlE!)Fhal7$Feu$\"3g'RN#Q,8[8!#77$Fju$\"3_'zz;GO\"*=#Febl 7$F_v$\"3e+I.x$Q?V$Febl7$Fdv$\"32#)=,s'*GRbFebl7$Fiv$\"3%GC4Nb*)GP)Feb l7$F^w$\"3IQB@zrI!G\"!#67$Fcw$\"3/$)\\U)R*yb=Fecl7$Fhw$\"3![\"Q#Q)ebYF Fecl7$F]x$\"3Ab))*3>uK#RFecl7$Fbx$\"3e:R#=Wxyi&Fecl7$Fgx$\"3!*4jpk`L?z Fecl7$F\\y$\"3P8>dH$*G?6!#57$Fay$\"3-DU\"Hj*))[:Fhdl7$Ffy$\"3qwF#p=6n8 #Fhdl7$F[z$\"3dz!y*>qS8HFhdl7$F`z$\"3;83\"eP%)[%QFhdl7$$\"3Q%3x^t:CE'F -$\"3;7)G1<#\\$\\%Fhdl7$Fez$\"3U-nK[M6S_Fhdl7$$\"3<](oC(f7FkF-$\"3c+*e 6lU;+'Fhdl7$Fjz$\"3[%>z#4)4D'oFhdl7$$\"39#H2e_/ye'F-$\"3N2n$3G#y-zFhdl 7$F_[l$\"3K/CY)*4_%3*Fhdl7$$\"3=,+Dxq2]nF-$\"3Azq@v`QO5!\"*7$Fd[l$\"3W F.`7h\\!=\"Fbgl7$$\"3%H1*y**p2soF-$\"3U%[*4J#R_E\"Fbgl7$$\"3Iv$f)**zr9 pF-$\"3)40To9yaN\"Fbgl7$$\"3k(oH****et&pF-$\"3!zr'yW;`^9Fbgl7$Fi[l$\"3 ;99RX[t`:Fbgl-F_\\l6&Fa\\lFe\\lFe\\lFb\\l-Fg\\l6#\"\"\"-F$6%7Y7$F($!3_ 8xHtziL9F-7$F>$!3P?z1XG7,5F-7$FH$!3\\411NH^MqF*7$FM$!3C12g)*=_BVF*7$FR $!3%\\sE)om&e4#F*7$FW$!3iAj&3_OHY#FY7$Fgn$\"3u&fvI.'3A7F*7$F\\o$\"319k S4qQ]DF*7$Fao$\"3uQzb2V#Gw$F*7$Ffo$\"3mAycCb`T[F*7$F[p$\"3OUz(\\#z*=%e F*7$F`p$\"3A9un$eKrk'F*7$Fep$\"3'=K$\\u?D#[(F*7$Fjp$\"3!p.@M)3-c#)F*7$ F_q$\"3)*))**oTo**[*)F*7$Fdq$\"314:Hk4@R&*F*7$Fiq$\"3]0im&R])>5F-7$F^r $\"3UI/aw(*[s5F-7$Fcr$\"3w7R;WpUI6F-7$Fhr$\"3%4km#*>_!z6F-7$F]s$\"3z** *y***\\\")H7F-7$Fbs$\"3g$ocm!)peF\"F-7$Fgs$\"3vR)[Q)Rw@8F-7$F\\t$\"3c1 ([^*38i8F-7$Fat$\"3H:L;A(>RS\"F-7$Fft$\"3kk*\\2NabW\"F-7$F[u$\"3Syx['3 T/[\"F-7$F`u$\"36Shj*>7o^\"F-7$Feu$\"3S\"zF8J#3`:F-7$Fju$\"3a%=SwqDue \"F-7$F_v$\"3WV+$H'zw>;F-7$Fdv$\"3g1FVH/,b;F-7$Fiv$\"3q2+H,$>mo\"F-7$F ^w$\"3!\\%Hq\">F8s\"F-7$Fcw$\"3-(>z(3Y8bF-7$Fgx$\"3lvrQSC85?F-7$F\\y$\"3!Hd'> [-ed@F-7$Fay$\"3)QPg(f%pGP#F-7$Ffy$\"3QH([T,7!3FF-7$F[z$\"3FH0='\\zP@$ F-7$F`z$\"3nL%eXYn?!RF-7$Fez$\"3YcBiFB?u]F-7$F^fl$\"3?^@M]&>gx&F-7$Fjz $\"3o*Hdh93jh'F-7$Fffl$\"3%erwZt%\\#p(F-7$F_[l$\"3[3&)fx%f'))*)F-7$F^g l$\"3E5>@;VBZ5Fj_l7$Fd[l$\"31Q]`+u]B7Fj_l7$Fggl$\"3+2^\"*)3c8L\"Fj_l7$ F\\hl$\"3g3(oxH2%\\9Fj_l7$Fahl$\"3#fUZt)z_y:Fj_l7$Fi[l$\"3`XwD(4]'> " 0 "" {MPLTEXT 1 0 42 "conv_plot( f, g, h, -0.8, 7, c1, c2, .15);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6%7U7$$!3U+++++++!)!#=$!3]+TV7zV4;!#<7$$!35,+]()4\"*\\r F*$!3Yp]um[Bb7F-7$$!3i+++v>#)*H'F*$!3I_[O'fT?%**F*7$$!3'***\\iqS;gbF*$ !3\"*Q:w+nn>\")F*7$$!3R++Dmh]?[F*$!3k$ \"3y&3XoYS%G=FO7$$\"3M***\\Px#yH\"y$[V/%F*7$$\"3y***\\( GU*Rj'F*$\"31!et\\`L')3&F*7$$\"3B,++XjkI$)F*$\"3q&4*G)H#*)fgF*7$$\"3a* ***\\_%e]#)*F*$\"3AoAYHihVoF*7$$\"37++]`7u]6F-$\"3s+9!z^D\"ewF*7$$\"3% )****\\_um>8F-$\"38T[kf$QUT)F*7$$\"3K++]Y\"fC[\"F-$\"3ehg0-m\\#4*F*7$$ \"3/+]P+'*GI;F-$\"3M]dQV&R4n*F*7$$\"3<++]3V21=F-$\"3DyCLjkyJ5F-7$$\"33 +++9q)\\&>F-$\"3/dHEcU\\$3\"F-7$$\"3A+]P\"*y?G@F-$\"3j@;\">Gg/9\"F-7$$ \"39+++>x`\"G#F-$\"3y)4/$H@J)=\"F-7$$\"3%***\\PT[w\\CF-$\"3M\\)*zmgIQ7 F-7$$\"3^+]7Wo&*4EF-$\"3cAtt\"oKF-$\"3!z.JwVt6X\"F-7$$\"3(***\\i'H?yT$F-$\"31*Q$*) Gkk&[\"F-7$$\"3E++vr2]zNF-$\"3hbOQ!***e@:F-7$$\"3'******H_Klu$F-$\"3'* [CJcVTd:F-7$$\"3x***\\Z]S*4RF-$\"3KcTdC=E\"f\"F-7$$\"3K+]PHh/oSF-$\"3% 34/XObHi\"F-7$$\"3k++DpafVUF-$\"3'=#yG@u+d;F-7$$\"3\")******eVL,WF-$\" 35,PcEgk'o\"F-7$$\"3w++]#Ha(pXF-$\"3S4\"4.%4N<1I4Rx(=F-7$$\"3v+++uJZ+d F-$\"3W4$))o9y@!>F-7$$\"31++DoZ!f'eF-$\"3XZ7*[#ycE>F-7$$\"3X+]P0l'*HgF -$\"33'>%GT>=]>F-7$$\"33,++.As!='F-$\"3)y!yPj**Rr>F-7$$\"3%4+]_C0NN'F- $\"3Mmxl.r<&*>F-7$$\"3`+++eW/3lF-$\"31')Q7S](f,#F-7$$\"3=+]P\"\\:Gn'F- $\"3V(\\Gki$oP?F-7$$\"3W+]i%RA0$oF-$\"35+I%[AH!e?F-7$$\"\"(\"\"!$\"3uN )z;aT%z?F--%'COLOURG6&%$RGBG$\"*++++\"!\")$Fa[lFa[lF[\\l-%*THICKNESSG6 #\"\"%-F$6%7Z7$F($!3[4_>)p$o'e\"F-7$F4$!3@(3?]\"3=M**F*7$F>$!3)>Xl\\NS &ylF*7$FC$!3Cq+WsSQ$z$F*7$FH$!3]rkkTcD/;F*7$FM$\"38'3XoYS%G=FO7$FS$\"3 mOa*o]gaf\"F*7$FX$\"3#)4&frkwT(GF*7$Fgn$\"36[_D@F[WSF*7$F\\o$\"3'f)py@ BR#4&F*7$Fao$\"3(y7.'eee7hF*7$Ffo$\"39&\\g*4&>x>(F*7$F[p$\"3\"eog\\O71 %)*F*7$F`p$\"3SX)[F-7$Fjp$\"37fAh]D@s7!#;7$ F_q$\"3m,9a:%=j(QFc_l7$Fdq$\"31&f/tbMZR*Fc_l7$Fiq$\"3+!=\\tb*>XC!#:7$F ^r$\"3O*H&*p5'el`F]`l7$Fcr$\"3#fOC\\7#y)>\"!#97$Fhr$\"31Sc$*\\(f9X#Fd` l7$F]s$\"34MS>%Qv!R\\Fd`l7$Fbs$\"39Ncz0N\")e!*Fd`l7$Fgs$\"31J2!Q%*QCo \"!#87$F\\t$\"31qNTt'3:4$Faal7$Fat$\"323g=mI&f6&Faal7$Fft$\"3^o?<4`V\"!#77$F`u$\"3?1ERn**\\I 8!#67$Fiv$\"3Sh/&>)4@@>F^cl7$F^w$\"3'oH'*=u5I$GF^cl7$Fcw$\"3Kz[7%>(yLS F^cl7$Fhw$\"3')3)HG(y@odF^cl7$F]x$\"3/;a5]GU%4)F^cl7$Fbx$\"3b_zWnLmT6! #57$Fgx$\"3'3uyf()yVd\"Fadl7$F\\y$\"3q)z)*>^\"[m@Fadl7$Fay$\"3G7383#Gr %HFadl7$Ffy$\"3cJ_$R#G]\")QFadl7$$\"3+,]7CP6niF-$\"3ir#)*\\V=7`%Fadl7$ F[z$\"3aP*>5!eOy_Fadl7$$\"3=,]i^[xIkF-$\"3FNokb$H(RgFadl7$F`z$\"3%oEo9 Px'**oFadl7$$\"3!3](ou*H/f'F-$\"3D;Ke0u%z$zFadl7$Fez$\"3'pN7c&RR;\"*Fa dl7$$\"3K+++V*o;v'F-$\"3h.0e386R5!\"*7$Fjz$\"3k;:Hb%*e#=\"F[gl7$$\"3m \\(ofz\"*G(oF-$\"3)\\dBv/6pE\"F[gl7$$\"3y*\\7t>h_\"pF-$\"3AkKl+Xmc8F[g l7$$\"3))\\il)fIw&pF-$\"3-ATi&zi@X\"F[gl7$F_[l$\"3;99RX[t`:F[gl-Fe[l6& Fg[lF[\\lF[\\lFh[l-F]\\l6#\"\"#-F$6%7Y7$F($!3?+'Q84s`H\"F-7$F4$!3+&egu ,')Q'*)F*7$F>$!3UN:kVya(>'F*7$FC$!3#3F_2\"HtfOF*7$FH$!3)Ri:!GRre:F*7$F M$\"3[irB+oax>FO7$FS$\"3>*R[*yyX+;F*7$FX$\"3w<=H?qmvGF*7$Fgn$\"3jHjP*y KZ/%F*7$F\\o$\"3U3%puN?()3&F*7$Fao$\"3SiP0Dz!*fgF*7$Ffo$\"3F$o;O/>O%oF *7$F[p$\"3eti\"p\"e7ewF*7$F`p$\"3m0B:z$QUT)F*7$Fep$\"3A2\\u-m\\#4*F*7$ Fjp$\"37\"H(RV&R4n*F*7$F_q$\"3qyCLjkyJ5F-7$Fdq$\"3#o&HEcU\\$3\"F-Fhq7$ F^r$\"3aMTIH@J)=\"F-7$Fcr$\"3Y)G4o11$Q7F-7$Fhr$\"3wDv3=ep$G\"F-7$F]s$ \"3L$R/)Hm&*G8F-7$Fbs$\"3;K=.#o*yo8F-7$Fgs$\"3R6sWhy/59F-7$F\\t$\"3YOE Q*Qv6X\"F-7$Fat$\"3C0^BPOl&[\"F-7$Fft$\"3S]p:^ah@:F-7$F[u$\"3uw!=he'\\ d:F-7$F`u$\"3oGMiZZ\\\"f\"F-7$Feu$\"34rzp_BaB;F-7$Fju$\"3#[r,\"H\"=&e; F-7$F_v$\"3eF*o%Q^'**o\"F-7$Fdv$\"3\">5&3Z;iCF-7$F]x$ \"3h*QP6l%eJ1)4X\"Fc_l7$Fjgl$\"3]]KW^yQz:F c_l7$F_[l$\"3`XwD(4]'>\"zV4;F-7$F[\\lF[\\lFahl-%*LINESTYLEG6#\"\"$-F$6%7$7$$FehlF a[l$\"3y4\"o')G7')4\"F-7$F\\dmFb[lF[cmFdcm-%)POLYGONSG6%7&7$$Fj[l!\"\" $!+7zV4;F[glFddm7$$\"#qFfdm$\"+U:Wz?F[gl7$FjdmFgdm-Fe[l6&Fg[l$\")viobF j[l$\")!\\DP\"Fj[l$\")%yg>%Fj[l-%&STYLEG6#%,PATCHNOGRIDG-Fadm6%7&7$Fed mF\\emFddmFidmFidm-Fe[l6&Fg[lFa[lFa[lFa[lFgem-%*AXESSTYLEG6#%&FRAMEG-% +AXESLABELSG6%Q\"x6\"Q\"yFifm%(DEFAULTG-%%VIEWG6$;FedmF_[l;$\"\"\"Fa[l F\\em" 1 2 0 1 10 0 2 9 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 433 "The red \+ curve is the original function. the blue graph is the series expanded \+ around x = 0, and the purple bar indicates the interval of convergence . The green graph is the series expanded about x = 2, and the yellow b ar indicates its interval of convergence. From this diagram, it appea rs the expansion about 2, has interval of convergence approaching (-1, 5), instead of (-1,1) for the other. Lets look at another expansion v alue." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "c1 := 0; c2 := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#c1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "g := unapply( convert(series( f(x), x = c1, 12), polynom), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR 6#%\"xG6\"6$%)operatorG%&arrowGF(,89$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.!\" \"*&#F.\"\"$F.)F-F7F.F.*&#F.\"\"%F.*$)F-F;F.F.F4*&#F.\"\"&F.)F-F@F.F.* &#F.\"\"'F.*$)F-FDF.F.F4*&#F.\"\"(F.)F-FIF.F.*&#F.\"\")F.*$)F-FMF.F.F4 *&#F.\"\"*F.)F-FRF.F.*&#F.\"#5F.*$)F-FVF.F.F4*&#F.\"#6F.)F-FenF.F.F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "h := unapply( convert (series( f(x), x = c2, 12), polynom), x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(,<-%#lnG6#\"\"& \"\"\"*&#F1F0F19$F1F1#\"\"%F0!\"\"*&#F1\"#]F1*$),&F4F1F6F7\"\"#F1F1F7* &#F1\"$v$F1)F=\"\"$F1F1*&#F1\"%+DF1*$)F=F6F1F1F7*&#F1\"&Dc\"F1)F=F0F1F 1*&#F1\"&]P*F1*$)F=\"\"'F1F1F7*&#F1\"'voaF1)F=\"\"(F1F1*&#F1\"(+]7$F1* $)F=\"\")F1F1F7*&#F1\")D\"yv\"F1)F=\"\"*F1F1*&#F1\")]il(*F1*$)F=\"#5F1 F1F7*&#F1\"*v$4r`F1)F=\"#6F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "conv_plot(f, g, h, -0.8, 12, c1, c2, .15.2 );" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6,-%'CURVESG6%7X7$ $!3U+++++++!)!#=$!3]+TV7zV4;!#<7$$!3enmmm5\\-tF*$!38059jjD58F-7$$!3iLL LL@)\\g'F*$!35&4c$pgF!3\"F-7$$!3l*******>tu!fF*$!3=t)y=9DU$*)F*7$$!3zm mmmU'*4_F*$!3?`'Hh@s/O(F*7$$!33+++!omh*RF*$!3qe_+6%p=5&F*7$$!3QLLL$4pB y#F*$!3aS/1KKegKF*7$$!3-+++gnK<9F*$!3n-N(eh'RG:F*7$$!38mmmmEWG_!#?$!3` h3F.)e@C&FT7$$\"3kKLL8q*ep#F*$\"3Q5$fXyPpQ#F*7$$\"3&emmm#o,JaF*$\"3,s1 S0Z%zL%F*7$$\"3)GLLL,Bo'zF*$\"3!))\\dD)zTfeF*7$$\"3%)*****RM]#f5F-$\"3 HyJ!*e+UBsF*7$$\"3LLL(zP^7#F- $\"3%*oGOx$y%R6F-7$$\"3Y+++;k@,CF-$\"34l'\\q8LTA\"F-7$$\"3#*******R'G% yEF-$\"3mz.*4m&[-8F-7$$\"39+++%)RdXHF-$\"3#R=#HSWfs8F-7$$\"3YLLL\\v;)= $F-$\"3Wq@@*HjAV\"F-7$$\"3]mmmiZjwMF-$\"3KU/6&fr))\\\"F-7$$\"39nmmIV+@ PF-$\"3-1N))e:-_:F-7$$\"3/+++KUE0SF-$\"3.W+)\\?!\\5;F-7$$\"31mmm5C)oD% F-$\"3UyIW`\"Q&f;F-7$$\"3G+++Ku%H`%F-$\"3yPv0j1s5K&F-$ \"3@a+&4tMS%=F-7$$\"3cLLL`?o$f&F-$\"3C0b\"f#>6')=F-7$$\"3;LLL\\\"*)e(e F-$\"3\"yC-g'4-G>F-7$$\"30+++c7b@hF-$\"3!)4nsub7j>F-7$$\"3aLLLhI(oQ'F- $\"3#>'oL9Xq**>F-7$$\"3c*****zWw4m'F-$\"35qwD\\%Rh.#F-7$$\"3\"******f8 L\"HpF-$\"3M*fp^rV02#F-7$$\"3q******>&)e)=(F-$\"3dn?8T:u-@F-7$$\"3w*** **z[pmZ(F-$\"3!p4,Oi,)F-$\"3sgFJBka)>#F-7$$\"3/KLLLrMi#)F-$\"3r`Bm'[dfA#F-7$$\"3V*****R5 Zh`)F-$\"3=3&yj`*3bAF-7$$\"3gkmmQ:x$z)F-$\"3)>ZXjiYg*F-$\"3=h7;$\\S5O#F-7$$\"3mLLLdJWn)*F-$\"3y)*zM`9x&Q#F-7$$\"3 BLL`k?*Q,\"!#;$\"3a8!*zR`W5CF-7$$\"3ommE;]\"3/\"F]z$\"3r5>^E!GVV#F-7$$ \"3!*****z7Wbl5F]z$\"3aQqid>ybCF-7$$\"3mmm1`&3R4\"F]z$\"3e0+J;v\")zCF- 7$$\"3=LL8M)o#>6F]z$\"3>nKzbk$3]#F-7$$\"3/++?$=3j9\"F]z$\"3&*H!=3#3xAD F-7$$\"3#*****R!H)=s6F]z$\"3cz5+uNKVDF-7$$\"#7\"\"!$\"3uO:Yd$\\\\c#F-- %'COLOURG6&%$RGBG$\"*++++\"!\")$Fa\\lFa\\lF[]l-%*THICKNESSG6#\"\"%-F$6 %7Z7$F($!3[4_>)p$o'e\"F-7$F>$!3U04\\Er#)ftF*7$FH$!3]O1^!*HegKF*FQ7$FX$ \"3uNq#H)y$pQ#F*7$Fgn$\"3fKwq%[5$QVF*7$F\\o$\"3Sy];^r%3*eF*7$Fao$\"3mo 'Ht[,a1)F*7$Ffo$\"32!*[)QclB+#F-7$F[p$\"3Cr55Ve\"[0\"F]z7$F`p$\"3G'F b`l7$Fcr$\"3z@Y$Gmo#H8!#77$Fhr$\"3<)>2&4jROIFi`l7$F]s$\"34xj$o+\\8,'Fi `l7$Fbs$\"3?%*\\3*\\l]@\"!#67$Fgs$\"3erR/\\&QLG#Fcal7$F\\t$\"3c'\\\"zW 7l_UFcal7$Fat$\"3m#[hA_61J(Fcal7$Fft$\"3'zdSNRiZF\"!#57$F[u$\"3GESk-G) z?#F`bl7$F`u$\"3'eQJOB%o'[$F`bl7$Feu$\"3A!>38JRdf&F`bl7$Fju$\"3U#z.oj, y$*)F`bl7$F_v$\"3YMSiaGG(Q\"!\"*7$Fdv$\"3=kk$R*[$*)3#F`cl7$Fiv$\"3YH;^ 7'o[B$F`cl7$F^w$\"3b0MF@&ySs%F`cl7$Fcw$\"3yg[h=Ta!)pF`cl7$Fhw$\"3%4%=9 H%*HH)*F`cl7$F]x$\"3(3N4**3>AT\"Fj\\l7$Fbx$\"3yXr681Fj\\l7$Fgx$\"3W j'yM0puu#Fj\\l7$F\\y$\"3W-L*3^[ix$Fj\\l7$Fay$\"39rTutv9=_Fj\\l7$Ffy$\" 3iJ[=RQ=jqFj\\l7$F[z$\"3w.7,!Qkga*Fj\\l7$Faz$\"3_(pD;x,oF\"!\"(7$Ffz$ \"3kgZs94&pl\"Fhel7$$\"3GLL$H[J(z5F]z$\"3.&=M#y')f=>Fhel7$F[[l$\"3z$>v #ojGDFhel7$F`[l$\"3!eF;(Hu&* eGFhel7$$\"3_mmm3&)yK6F]z$\"3W/][G:5mKFhel7$Fe[l$\"3UK@+8`HDPFhel7$$\" 3!*****zO#[#f6F]z$\"3\"om]vM2!>UFhel7$Fj[l$\"3m,Xj.w]rZFhel7$$\"3#)*** *z<79z6F]z$\"3+aJIE*3[4&Fhel7$$\"3))****>XT4'=\"F]z$\"3^ab)*e\\!zV&Fhe l7$$\"3%*****fsq/$>\"F]z$\"3W8s'z&*z=!eFhel7$F_\\l$\"3))>0[d1(y='Fhel- Fe\\l6&Fg\\lF[]lF[]lFh\\l-F]]l6#\"\"#-F$6%7Y7$F($!30O\\[y9765F-7$F>$!3 EFMs\\.ApcF*7$FH$!3\\0I&fkx#3EF*7$FR$\"3#Q!HEo2c'z\"!#>7$FX$\"37ppD>R_ oCF*7$Fgn$\"3c8d!=5feO%F*7$F\\o$\"3?]q_wgFpeF*7$Fao$\"3u(*G^EscEsF*7$F fo$\"3!*QM&)\\T*HY)F*7$F[p$\"3%3'[5:D,h&*F*7$F`p$\"3s0],L0wd5F-7$Fep$ \"3Fpnkj$z%R6F-7$Fjp$\"3/ZfDsK8C7F-7$F_q$\"3L*\\8Qn&[-8F-7$Fdq$\"3)f,$ 4TWfs8F-7$Fiq$\"3*Q3X#*HjAV\"F-7$F^r$\"3)>g5^fr))\\\"F-FbrFgr7$F]s$\"3 ?yIW`\"Q&f;F-7$Fbs$\"3a+x0j1s56')=F-7$F[u$\"3'>`5wW@!G> F-7$F`u$\"3kJ+/:w7j>F-7$Feu$\"3Y@(Q1i7(**>F-7$Fju$\"3z(46pJoh.#F-7$F_v $\"3w6Fca@jq?F-7$Fdv$\"3=x\"G>1zH5#F-7$Fiv$\"3>(*3]\"fmz8#F-7$F^w$\"33 $Q(\\93*)o@F-7$Fcw$\"3\\kZBp'f>?#F-7$Fhw$\"3F+:PjW$GB#F-7$F]x$\"3a(zfR \"y?pAF-7$Fbx$\"3NZ#=f>_%3BF-7$Fgx$\"3'>+!=a%4!fBF-7$F\\y$\"3KZb8/)oXU #F-7$Fay$\"3Fq\\:h[e@DF-7$Ffy$\"33vDV(fM!fEF-7$F[z$\"3!oDOE,_'pGF-7$Fa z$\"3#3*\\99&ya=$F-7$Ffz$\"3Hk;c?l-:OF-7$F[[l$\"3uo;hVJc[VF-7$Fefl$\"3 Y_:BMn>*y%F-7$F`[l$\"3%yPbogv!=`F-7$F]gl$\"3OuULZCH(*fF-7$Fe[l$\"3R%G2 yQqx\"oF-7$Fegl$\"3)Hp*4#\\W(fxF-7$Fj[l$\"3w\"Q6%p'*=#)))F-7$F]hl$\"3) 4R8yZx/d*F-7$Fbhl$\"3H]-dUx]K5F]z7$Fghl$\"3>dC\"=iu^6\"F]z7$F_\\l$\"3H &y.H7oc?\"F]z-Fe\\l6&Fg\\lF[]lFh\\lF[]lF`il-F$6%7$7$F[]l$!3(******>\"z V4;F-7$F[]lF[]lF^il-%*LINESTYLEG6#\"\"$-F$6%7$7$$F_]lFa\\l$\"3G+TV7zV4 ;F-7$FgdmFb\\lFfcmF_dm-%)POLYGONSG6%7&7$$Fj\\l!\"\"$!+7zV4;F`clF_em7$$ \"$?\"Faem$\"+d$\\\\c#F`cl7$FeemFbem-Fe\\l6&Fg\\l$\")viobFj\\l$\")!\\D P\"Fj\\l$\")%yg>%Fj\\l-%&STYLEG6#%,PATCHNOGRIDG-F\\em6%7&7$F`emFgemF_e mFdemFdem-Fe\\l6&Fg\\lFa\\lFa\\lFa\\lFbfm-%*AXESSTYLEG6#%&FRAMEG-%+AXE SLABELSG6%Q\"x6\"Q\"yFdgm%(DEFAULTG-%%VIEWG6$;F`emF_\\l;$\"\"\"Fa\\lFg em" 1 2 0 1 10 0 2 9 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "The expansion \+ about x = 4 seems to have an interval of convergence approaching (-1, 9 ). Its a little hard to see where it cuts off on the right. The fol lowing plot helps to demonstrate this." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot( \{f(x), g(x), h(x) , [[9,0], [9,h(9)]] \}, x = -.5..12, y = -1..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7$7$$\"\"*\"\"!$F*F*7 $F($\"3!>6yR#>)fM#!#<-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-F$6$7Z7$$!3+++++ +++]!#=$!35*Q`NX#4JpF=7$$!3]mm;HdNvAF=$!3'*\\L@$=$p\"e#F=7$$\"3.)pm\"H #oU`*!#?$\"3KZW?RR5*[*FH7$$\"3iLLe*)4WhFF=$\"34:sC=7VQCF=7$$\"3YNL3F>@ XaF=$\"31fd%oj;vM%F=7$$\"3[m;zptA;\")F=$\"3=#ozv?+7)fF=7$$\"3GLe*)f+Ef 5F/$\"3av[(QaPb1)F=7$$\"3/+D1kTn:8F/$\"3#yVbt`0S&=F/7$$\"3WLeR(*z&3e\" F/$\"3,dh')\\\"*=A#*F/7$$\"3;+D\"GQ\">X=F/$\"3-$*\\$HPi!>\\!#;7$$\"36n mTg24<@F/$\"3nNDq0^#[I#!#:7$$\"3wL$3-))zlN#F/$\"3L='z&o\\hMxF^p7$$\"3l ++D1z=EEF/$\"3'o[$f%G$=HE!#97$$\"33++voH!p*GF/$\"3Oh'o\"HV*>&zFip7$$\" 3c++v$4(ydJF/$\"3R!*=n!f&\\+@!#87$$\"3mLeRs[p%R$F/$\"3rnfypY)*RZFdq7$$ \"3YnmTN6SwOF/$\"3oMiGi=$4;\"!#77$$\"3lmmm\"HV]\"RF/$\"3)*pw[#He;N#F_r 7$$\"3U+Dc^3k#>%F/$\"3[\"QC<,N&p]F_r7$$\"30nm;HaH#\\gFds7$$\"37n\"zW<5&yaF/$\"3d+f!G+D0, \"!#57$$\"3*QL3-8IQu&F/$\"34%e)4\"35Lr\"Fdt7$$\"3(R$eR(*HU>gF/$\"3]Zv* 4$z:!*GFdt7$$\"3k+vV)RF$fiF/$\"3m+(\\asZ)oWFdt7$$\"3nL$eRsI%=lF/$\"3*> 263x'QBqFdt7$$\"3M++]7)4hy'F/$\"3a'oKQ:3(*4\"!\"*7$$\"3O+]7y:)z/(F/$\" 3]/S/Z6`w;F^v7$$\"3C+DcwfN,tF/$\"3')QTF`>7%[#F^v7$$\"3c**\\(o/&o#e(F/$ \"3u$yAR$Q`$y$F^v7$$\"3nmm;HFaz7$$\"3smTNr]R'4\"Fho$\"3kI-OL9$QF#Faz7$$\"33]P4@!y(36Fho$\"3?h+ H(o[ad#Faz7$$\"3WLL$3(4;@6Fho$\"3c.Q$*4C,8HFaz7$$\"3*=z%*zxjV8\"Fho$\" 3E<(H33=oJ$Faz7$$\"3;]i:&emv9\"Fho$\"3as&)HD**)3x$Faz7$$\"35++DcL?g6Fh o$\"3kZ0h$f0xD%Faz7$$\"3=]PMF,%G<\"Fho$\"3bED([[/5![Faz7$$\"3u7y]&4I'z 6Fho$\"3SCy7MNF=^Faz7$$\"35v=nj+U'=\"Fho$\"3WP_^;7\\aaFaz7$$\"3YPf$=.5 K>\"Fho$\"3w+D6@,n5eFaz7$$\"#7F*$\"3))>0[d1(y='Faz-F16&F3F+F4F+-F$6$7Y 7$F;$!31-^0%=z$y`F=7$FA$!3Ep(pZP1S/#F=7$FF$\"3:O)Ri\\zQ9$!#>7$FL$\"3od 4#='Q)z^#F=7$FQ$\"3C[#[el&*[P%F=7$FV$\"3#GC6%oa[^fF=7$Fen$\"3V6!o1F9mA (F=7$Fjn$\"33PLa`1'zR)F=7$F_o$\"3KbF'Gon9[*F=7$Fdo$\"3[\"y%)*GgjX5F/7$ Fjo$\"3'ff?*e6!p8\"F/7$F`p$\"38hg;`F#4@\"F/7$Fep$\"3=2`e`>=)G\"F/7$F[q $\"3MJxZG@=g8F/7$F`q$\"3I7?<%)H)\\U\"F/7$Ffq$\"3GN>l-\")R![\"F/7$F[r$ \"3;kw%f#)GDa\"F/7$Far$\"3Sr3jh0I#f\"F/7$Ffr$\"37bJO-CCZ;F/7$F[s$\"3)G c&yiyZ$p\"F/7$F`s$\"3%[D,Y2h=u\"F/7$Ffs$\"3'G&e9uW&ey\"F/7$F[t$\"37+'p Oy#yH=F/7$F`t$\"3s*y^=g!\\o=F/7$Fft$\"3iGUNO#G'3>F/7$F[u$\"3-(HGZ;#o[> F/7$F`u$\"3![iHJW\"H#)>F/7$Feu$\"3v`n+8EP.8K)QYDF/7$Fhy$\"3QKhSQOd\"p#F/7$F]z$\"3t0,BEsS7HF/7$Fcz$ \"3uMP&=!>TSKF/7$Fhz$\"3oQGg*pY@o$F/7$Fb[l$\"37\\A;f!)pGWF/7$Fg[l$\"3! y#=Cx5zt[F/7$F\\\\l$\"3cK$*RVg^0aF/7$Fa\\l$\"3!eR7*\\2:&3'F/7$Ff\\l$\" 3j4+\\69B-pF/7$F[]l$\"3PR\\w]52OyF/7$F`]l$\"3#[P?p&o/W*)F/7$Fe]l$\"3Lw Z+xJI@'*F/7$Fj]l$\"3+O`,+r@O5Fho7$F_^l$\"3R.7o%[.s6\"Fho7$Fd^l$\"3H&y. H7oc?\"Fho-F16&F3F4F4F+-F$6$7U7$F;$!3'GX*f0=ZJpF=7$$!3(HL$ekynPOF=$!3f c&*[mm\">_%F=7$FA$!3O;f`.Kp\"e#F=7$$!3w*\\PM_1+4\"F=$!3ILYfOe6a6F=7$FF $\"3OeX?RR5*[*FH7$FL$\"3$y!fk(3J%QCF=7$FQ$\"3A#oioaRrM%F=7$FV$\"3$**) \\$R$)HA%fF=7$Fen$\"3#Ho/0CnMA(F=7$Fjn$\"37`#4sg3qR)F=7$F_o$\"3'=G)y2! >7[*F=7$Fdo$\"35[r>N.jX5F/7$Fjo$\"3rwjB=,!p8\"F/7$F`p$\"3?'GvFcA4@\"F/ 7$Fep$\"3YZ6!H$>=)G\"F/7$F[q$\"3'*[e3F@=g8F/7$F`q$\"3gA07%)H)\\U\"F/7$ Ffq$\"3O05l-\")R![\"F/7$F[r$\"3]jw%f#)GDa\"F/Fabl7$Ffr$\"3MbJO-CCZ;F/7 $F[s$\"37ZbyiyZ$p\"F/7$F`s$\"3Lpkfu5'=u\"F/7$Ffs$\"3=\\w&RZaey\"F/7$F[ t$\"3Gs7D!y#yH=F/7$F`t$\"3u5Zes0\\o=F/7$Fft$\"31ww4K!G'3>F/7$F[u$\"3wT kb?5o[>F/7$F`u$\"3k'e,(zrG#)>F/7$Feu$\"3:!HgeVdt,#F/7$Fju$\"3)>.HHNTB0 #F/7$F`v$\"35\"RaVK@a3#F/7$Fev$\"3b)>H@()=k6#F/7$Fjv$\"3#ezpq!ou\\@F/7 $F_w$\"3)[)Rd(\\u(y@F/7$Fdw$\"3ruM(*\\\\')3AF/7$Fiw$\"3Hckh@3PNAF/7$F_ x$\"3S-Dg*zmNE#F/7$Fdx$\"33$[#\\t,R*G#F/7$Fix$\"3x#>r5k)o:BF/7$F^y$\"3 zw:R@%[2M#F/7$Fcy$\"3w$\\fZtFjO#F/7$Fhy$\"3?^G$G8g.R#F/7$F]z$\"3\"\\`c eg`VT#F/7$Fcz$\"3&=pj*pIfPCF/7$Fhz$\"3KP-)H!=[eCF/7$Fb[l$\"3]kc>@!)*=[ #F/7$F\\\\l$\"3MYun9rQ-DF/7$Ff\\l$\"39Wz3:+yBDF/7$F`]l$\"3EDUO