{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 } {CSTYLE "" -1 256 "" 1 14 0 0 138 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 36 0 0 0 0 0 0 0 0 0 0 1 }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 2 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 29 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 25 "Summing series with Map le" }}{PARA 0 "" 0 "" {TEXT -1 122 "The basic command for summing a se ries has the structure sum(expression,limits). For example if we wish \+ to sum the series " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "s = 1+1/(2^3)+1/( 3^3);" "6#/%\"sG,(\"\"\"F&*&F&F&*$\"\"#\"\"$!\"\"F&*&F&F&*$F*F*F+F&" } {TEXT -1 3 "..." }{XPPEDIT 18 0 "1/(1000^3);" "6#*&\"\"\"F$*$\"%+5\"\" $!\"\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "s:=sum(1/x^3,x=1. .1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&-%$PsiG6$\"\"#\"%,5 #\"\"\"F)-%%ZetaG6#\"\"$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "sum attempts to evaluate the series analytically (" }{XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 25 " is the digamma function " }{XPPEDIT 18 0 "ze ta" "6#%%zetaG" }{TEXT -1 132 " the Riemann zeta function). If neither of these function mean anything to you, don't worry. If you want a nu mber we can apply evalf" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e valf(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.k0-7!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "We can use infinity as the upper limit e. g. the harmonic series mentioned in class:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "s1:=-sum((-1)^n/n,n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G-%#lnG6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The result of a summation can also used to create a funct ion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x->sum(x^n/n!,n=1 ..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operator G%&arrowGF(-%$sumG6$*&)9$%\"nG\"\"\"-%*factorialG6#F2!\"\"/F2;F3\"\"'F (F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "This function can then \+ be used for further manipulations e.g. we can plot it. In order to cre ate anything but the simplest plots we need to read in the plot librar y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot(f(x),x=1..5,axes=boxed ,color=blue, labels=[x,\"f(x) \"],title=\"Partial sum of power series for exponetial\",tickmarks=[3,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6)-%'CURVESG6#7S7$$\"\"\"\"\"!$\"3nbbbbb0=F-7$$\"3WLL$e'40j6F-$\"3Gg$z,>9!*>#F-7$$ \"3ommm6hO[7F-$\"3)R;*4ly`$[#F-7$$\"3xmmm\"yYUL\"F-$\"3sk7:8'\\`z#F-7$ $\"3CLL$eF>(>9F-$\"3%oSe**4oJ8$F-7$$\"3kmm;>K'*)\\\"F-$\"3Or2@,c\"HZ$F -7$$\"3/++]Kd,\"e\"F-$\"33%>tF&3#Q&QF-7$$\"3gmm;fX(em\"F-$\"3s#eab/P9G %F-7$$\"3!*****\\U7Y]P \\ci_F-7$$\"3ommmhb59>F-$\"3+#=;_4alv&F-7$$\"3#*******H,Q+?F-$\"3%*)Rf $R$=$ejF-7$$\"3)*******\\*3q3#F-$\"3VfOk4\\-:qF-7$$\"3?+++q=\\q@F-$\"3 q)faa15;q(F-7$$\"3mmm;fBIYAF-$\"3a'44vOTUP)F-7$$\"30LLLj$[kL#F-$\"30p: !)y^eR#*F-7$$\"3?LLL`Q\"GT#F-$\"3#\\LsyYSK+\"!#;7$$\"3o****\\s]k,DF-$ \"3'\\:6a2()G5\"F]q7$$\"3#HLLLvv-e#F-$\"3I$y,A+J#)>\"F]q7$$\"33++]sgam EF-$\"3u5rK_P768F]q7$$\"3!)****\\G_)HF]q7$$\"3k*****\\@fke$F-$\"3sCDIU+1] KF]q7$$\"3/LLL`4NnOF-$\"31FFi\"f&o0NF]q7$$\"3#*******\\,s`PF-$\"3'HozQ ih#)z$F]q7$$\"3[mm;zM)>$QF-$\"3_GBx9s&>3%F]q7$$\"3$*******pfa^F]q7$$\"3;LLL$)G[kTF-$\"3#\\;*R\\!y)3bF]q7$$\"3#)****\\ 7yh]UF-$\"35W#*3J+7WfF]q7$$\"3Kmmm')fdLVF-$\"3I&p$))=A$=R'F]q7$$\"36mm m,FT=WF-$\"3EDUWSYH!)oF]q7$$\"3FLL$e#pa-XF-$\"3#f9!QA.,(R(F]q7$$\"3!** *****Rv&)zXF-$\"3O``+$)3s,zF]q7$$\"3%GLL$GUYoYF-$\"3\")H'GY:Et^)F]q7$$ \"33mmm1^rZZF-$\"35Cf3z,L.\"*F]q7$$\"34++]sI@K[F-$\"3U()3Zk0%pw*F]q7$$ \"34++]2%)38\\F-$\"31nx;m`:W5!#:7$$\"\"&F*$\"3dbbbb0=@6Fdz-%&TITLEG6#Q KPartial~sum~of~power~series~for~exponetial6\"-%'COLOURG6&%$RGBG$F*F*F c[l$\"*++++\"!\")-%+AXESLABELSG6$%\"xGQ'f(x)~~F^[l-%*AXESSTYLEG6#%$BOX G-%*AXESTICKSG6$\"\"$Fgz-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 19 "or differentiate it" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\" \"\"F$%\"xGF$*&#F$\"\"#F$)F%F(F$F$*&#F$\"\"'F$)F%\"\"$F$F$*&#F$\"#CF$) F%\"\"%F$F$*&#F$\"$?\"F$)F%\"\"&F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Taylor expansions can carried o ut by the command series(expression,expansion point,order)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(exp(x),x=2,5);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#+/,&%\"xG\"\"\"\"\"#!\"\"-%$expG6#F'\"\"!F)F&,$F )#F&F'F',$F)#F&\"\"'\"\"$,$F)#F&\"#C\"\"%-%\"OG6#F&\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "If the order is not specified Maple uses \+ a default value" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(s in(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"F%#!\"\"\" \"'\"\"$#F%\"$?\"\"\"&-%\"OG6#F%F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We can also make multivariable Taylor expansions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "mtaylor(exp(x+y),[x,y],5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,@\"\"\"F$%\"xGF$%\"yGF$*&#F$\"\"#F$)F %F)F$F$*&F&F$F%F$F$*&F(F$)F&F)F$F$*&#F$\"\"'F$)F%\"\"$F$F$*(F(F$F&F$F* F$F$*(F(F$F-F$F%F$F$*&F/F$)F&F2F$F$*&#F$\"#CF$)F%\"\"%F$F$*(F/F$F&F$F1 F$F$*(#F$F;F$F-F$F*F$F$*(F/F$F6F$F%F$F$*&F8F$)F&F;F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "If you are using an earlier version of M aple than Maple 6 you must, first request mtaylor as a library functi on, that is you must first write" }}{PARA 0 "" 0 "" {TEXT -1 18 ">read lib(mtaylor);" }}}}{MARK "13 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }