{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 266 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "Red Emphasis" -1 267 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 268 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Em phasis" -1 271 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{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 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 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 "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 52 "Levin's u transformation for summ ing infinite series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Na naimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 7.2.2008 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "load procedures for series" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 8 "series.m " }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 6 "le vinU" }{TEXT -1 5 " and " }{TEXT 0 8 "levinsum" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the one that follows, where t he file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/series.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Levin's u-t ransform" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 29 "The following description of " }{TEXT 266 19 "Levin' s u-transform" }{TEXT -1 23 " is due to Dirk Laurie." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Suppose that we have an infinite series " }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)" "6#-%$SumG 6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 24 "Let the partial sums be " }{XPPEDIT 18 0 "s[n]=Sum(a[i] ,i=1..n)" "6#/&%\"sG6#%\"nG-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'" }{TEXT -1 10 ", and let " }{XPPEDIT 18 0 "b[n]" "6#&%\"bG6#%\"nG" }{TEXT -1 51 " be an auxiliary sequence which tends to zero.\nThen" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "s[n] = ``(s[n]/b[n])/``(1/b[n] );" "6#/&%\"sG6#%\"nG*&-%!G6#*&&F%6#F'\"\"\"&%\"bG6#F'!\"\"F/-F*6#*&F/ F/&F16#F'F3F3" }{TEXT -1 15 " -------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(` `(s[n]/b[n])/``(1/b[n]),n = infinity);" "6#-%&LimitG6$*&-%!G6#*&&%\"sG 6#%\"nG\"\"\"&%\"bG6#F.!\"\"F/-F(6#*&F/F/&F16#F.F3F3/F.%)infinityG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "has the indeterminate fo rm [" }{XPPEDIT 18 0 "infinity/infinity" "6#*&%)infinityG\"\"\"F$!\"\" " }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Regard the numerator and denominator of (i) as functions \+ of " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 31 ", an d apply L'Hopital's rule. " }}{PARA 0 "" 0 "" {TEXT -1 138 "You can't do this directly, because the functions are only defined at certain i ntegers, so use divided differences instead of derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The u-transform is obtained using the auxiliary sequence" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n] = n*a[n];" "6#/&%\"bG6#%\"nG*&F'\"\"\"&% \"aG6#F'F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedu re " }{TEXT 0 6 "levinU" }{TEXT -1 50 " in the next section is based o n this description." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 266 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 399 "When floating point arithmetic is used there can be a significant bui ld-up of round-off error.\nThe decision concerning which term of the t ransform provides the best approximation to the sum of the series can \+ be hard to make and is a trade-off between accuracy of approximation a nd build-up of round-off error.\nIn a doubtful situation, increasing t he precision of the computation can provide a guide." }}{PARA 15 "" 0 "" {TEXT -1 89 "The method can break down for convergent series where \+ the terms tend to zero very slowly." }}{PARA 15 "" 0 "" {TEXT -1 190 " For numerical work it is desirable to try to prevent the numerators an d denominators in the quotient sequence of divided differences from gr owing too large by dividing by suitable constants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "A procedure for \+ constructing Levin's u transform: " }{TEXT 0 6 "levinU" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "levinU: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 263 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 14 " levinU( aa) " }{TEXT 265 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 9 " aa - " }{TEXT -1 31 " a list o f terms of a sequence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The pro cedure " }{TEXT 0 6 "levinU" }{TEXT -1 60 " constructs the Levin u-tr ansform of a sequence or series.\n" }}{PARA 0 "" 0 "" {TEXT 264 8 "Opt ions:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 196 "mode=series/se quence\nThe procedure constructs the u-transform of the series or sequ ence whose terms are given depending on whether the option \"mode=seri es\" or \"mode=series\" is chosen respectively." }}{PARA 0 "" 0 "" {TEXT -1 29 "The default is \"mode=series\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "info=true/false\nThis option al lows the progress of the procedure to be monitored." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 100 ": \+ Exact results are given when the input is exact, and symbolic sequence s and series can be handled." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the procedure activ e, open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "levinU: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3384 "levinU \+ := proc(ff::list)\n local Options,n,i,j,x,s,a,b,num,den,lev,md,\n \+ prntflg,temp;\n \n # Get the options \"mode\" and \"info\".\n # Se t the default values to start with.\n md := 'series';\n prntflg := false;\n if nargs>1 then\n Options := [args[2..nargs]];\n \+ if not type(Options,list(equation)) then\n error \"each optio nal argument must be an equation\"\n end if;\n if hasoption( Options,'mode','md','Options') then\n if not (md='sequence' or md='series') then\n error \"\\\"mode\\\" must be 'sequence ' or 'series'\"\n end if;\n end if;\n if hasoption(O ptions,'info','prntflg','Options') then\n if prntflg<>true the n prntflg := false end if;\n end if;\n if nops(Options)>0 th en\n error \"%1 is not a valid option for %2\",op(1,Options),p rocname;\n end if;\n end if;\n n := nops(ff);\n \n if md='s eries' then\n a := convert(ff,array);\n \n # Construct \+ sequence s of partial sums\n s := array(1..n);\n s[1] := a[1 ];\n for i from 2 to n do\n s[i] := s[i-1] + a[i];\n \+ end do;\n if prntflg then\n print(``);\n print(` sequence of partial sums ... `,convert(s,list));\n end if;\n el se\n s := convert(ff,array);\n\n # Construct sequence of dif ferences\n a := array(1..n);\n a[1] := s[1];\n for i fr om 2 to n do\n a[i] := s[i] - s[i-1];\n end do;\n if prntflg then\n print(``);\n print(`sequence of differ ences ... `,convert(a,list));\n end if;\n end if;\n\n # seque nces a and s are to be considered as functions of 1/index.\n for i f rom 1 to n do x[i] := 1/i end do;\n\n # Construct auxiliary sequence b[i]=a[i]*i\n b := array(1..n):\n for i from 1 to n do\n if \+ a[i]=0 then\n if md='series' then\n error \"terms o f initial seqence must all be non-zero\"\n else\n e rror \"consecutive terms of initial seqence must be distinct\"\n \+ end if;\n end if;\n b[i] := a[i]*i;\n end do;\n if pr ntflg then\n print(``);\n print(`auxiliary sequence ... `,co nvert(b,list));\n end if;\n\n # Construct quotient sequence with n um=s[i]/b[i] & den=1/b[i]\n num := array(1..n):\n den := array(1.. n):\n\n for i from 1 to n do\n num[i] := s[i]/b[i];\n den[ i] := 1/b[i];\n end do:\n if prntflg then\n print(``);\n \+ print(`INITIAL QUOTIENT SEQUENCE:`);\n print(`numerator sequence ... `,convert(num,list));\n print(`denominator sequence ... `,co nvert(den,list));\n end if;\n \n # Construct sequences of divide d differences for num & denom\n\011 for i from 2 to n\n\011 do\n\011 for j from n by -1 to i\n\011 do\n\011 temp := x[j] - \+ x[j-i+1];\n num[j] := (num[j] - num[j-1])/temp;\n den[ j] := (den[j] - den[j-1])/temp;\n\011 end do;\n end do;\n\n # \+ Form sequence of quotients\n lev := array(1..n):\n for i from 1 to n do\n if den[i]<>0 then\n # Divide num & den by i^i to \+ avoid them growing large. \n num[i] := num[i]/i^i;\n d en[i] := den[i]/i^i;\n lev[i] := num[i]/den[i];\n else\n \+ lev[i] := infinity\n end if;\n end do:\n\n if prntflg then\n print(``);\n print(`DIVIDED DIFFERENCES:`);\n p rint(`numerator sequence ... `,convert(num,list));\n print(`denom inator sequence ... `,convert(den,list));\n print(``);\n end if ;\n convert(lev,list);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "Examples are given in the following sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "The first few terms of Levin's \+ u-transform of a general series" }}{PARA 0 "" 0 "" {TEXT -1 23 "Activa te the procedure " }{TEXT 0 6 "levinU" }{TEXT -1 25 " in the previous \+ section." }}{PARA 0 "" 0 "" {TEXT -1 97 "Then you can execute the foll owing commands to see a Levin u transform of the finite series with " }{TEXT 270 1 "n" }{TEXT -1 7 " terms:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[0],a[1],` . . . `,a[n-1];" "6&&%\"aG6#\"\"!&F$6# \"\"\"%(~.~.~.~G&F$6#,&%\"nGF)F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "n := 4:\n a := array(0..n-1):\naa := [seq(a[i],i=0..n-1)];\nlevinU(aa);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG7&&%\"aG6#\"\"!&F'6#\"\"\"&F'6# \"\"#&F'6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%\"aG6#\"\"!*&,& *&#\"\"\"\"\"%F,*&,&F$F,&F%6#F,F,F,F0!\"\"F,F2#F,\"\"#F,F,,&*&#F,F-F,* &F,F,F0F2F,F2*&F3F,*&F,F,F$F2F,F,F2*&,(*&#F,\"\"*F,*&,(F$F,F0F,&F%6#F4 F,F,FBF2F,F,*&#F4F?F,F.F,F2F>F,F,,(*&F>F,*&F,F,FBF2F,F,*&#F4F?F,F8F,F2 *&F>F,F:F,F,F2*&,**&#F,\"#;F,*&,*F$F,F0F,FBF,&F%6#\"\"$F,F,FSF2F,F2*&# F?\"#kF,F@F,F,*&#FU\"#KF,F.F,F2#F,FXF,F,,**&#F,FPF,*&F,F,FSF2F,F2*&FWF ,FHF,F,*&#FUFenF,F8F,F2*&FfnF,F:F,F,F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Alternatively, once Maple has perfor med a u-transform in the calculation of a sum, the procedure " }{TEXT 0 19 "`evalf/Sum1/levinu`" }{TEXT -1 29 " is available. The procedure \+ " }{TEXT 0 6 "levinu" }{TEXT -1 45 " is taken from this built-in Maple procedure." }}{PARA 0 "" 0 "" {TEXT -1 84 "[ This procedure does not \+ reside in the Maple library, and so cannot be loaded with " }{TEXT 271 28 "readlib(`evalf/Sum1/levinu`)" }{TEXT -1 3 ". ]" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 36 ": The input array for the proced ure " }{TEXT 0 19 "`evalf/Sum1/levinu`" }{TEXT -1 27 " must have initi al index 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 123 "evalf(Sum(1/n^2,n=1..infinity)):\nn := 4:\na := ar ray(0..n-1):\n[seq(a[i],i=0..n-1)];\n`evalf/Sum1/levinu`(a):\nconvert( %,list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%\"aG6#\"\"!&F%6#\"\"\" &F%6#\"\"#&F%6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%\"aG6#\"\" !*&,&*&#\"\"\"\"\"%F,*&,&F$F,&F%6#F,F,F,F0!\"\"F,F,#F,\"\"#F2F,,&*&F+F ,*&F,F,F0F2F,F,*&#F,F4F,*&F,F,F$F2F,F2F2*&,(*&#F,\"\"*F,*&,(F$F,F0F,&F %6#F4F,F,FBF2F,F,*&#F4F?F,F.F,F2F>F,F,,(*&F>F,*&F,F,FBF2F,F,*&#F4F?F,F 7F,F2*&F>F,F:F,F,F2*&,**&#F,\"#;F,*&,*F$F,F0F,FBF,&F%6#\"\"$F,F,FSF2F, F,*&#F?\"#kF,F@F,F2*&#FU\"#KF,F.F,F,#F,FXF2F,,**&FOF,*&F,F,FSF2F,F,*&# F?FXF,FHF,F2*&FZF,F7F,F,*&#F,FXF,F:F,F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 ";\023" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 6 "levinU" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Geometric series" }}{PARA 0 "" 0 "" {TEXT -1 78 "We construct t he first few terms of Levin's u-transform of a geometric series:" }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(a*r^i,i = 0 .. i nfinity) = a+a*r+a*r^2+a*r^3+` . . . `+a*r^(n-1)+` . . . `;" "6#/-%$Su mG6$*&%\"aG\"\"\")%\"rG%\"iGF)/F,;\"\"!%)infinityG,0F(F)*&F(F)F+F)F)*& F(F)*$F+\"\"#F)F)*&F(F)*$F+\"\"$F)F)%(~.~.~.~GF)*&F(F))F+,&%\"nGF)F)! \"\"F)F)F9F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "n := 4;\na := 'a': r := 'r' :\naa := [seq(a*r^i,i=0..n-1)];\nbb := levinU(aa);\nfor i from 1 to n \+ do bb[i] := simplify(bb[i]) end do:\nbb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG7&%\"aG*&F& \"\"\"%\"rGF(*&F&F()F)\"\"#F(*&F&F()F)\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG7&%\"aG*&,&*(,&F&\"\"\"*&F&F+%\"rGF+F+F+F&!\"\"F- F.#F.\"\"%#F+\"\"#F+F+,&*&F+F+*&F&F+F-F+F.F/*&F1F+F&F.F+F.*&,(*(,(F&F+ F,F+*&F&F+)F-F2F+F+F+F&F.F-!\"##F+\"\"**&#F2F?F+F)F+F.F>F+F+,(*&F+F+*& F&F+F*&#F2F?F+F4F+F.*&F>F+F&F.F+F.*&,**(,*F&F+F,F+F;F+*&F&F+)F- \"\"$F+F+F+F&F.F-!\"$#F.\"#;**#F?\"#kF+F:F+F&F.F-F=F+*&#FN\"#KF+F)F+F. #F+FTF+F+,**&F+F+*&F&F+FMF+F.FP*(FSF+F&F.F-F=F+*&#FNFWF+F4F+F.*&FXF+F& F.F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&%\"aG*(F$\"\"\",&!\"\"F&% \"rGF&F&,&F(F&*&\"\"#F&F)F&F&F(,$*&F$F&F'F(F(F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "From the third term onwa rds the terms of the u-transform become constant and equal to the exac t expression for the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 18 "Exponential series" }}{PARA 0 "" 0 "" {TEXT -1 86 "We can construct the first few terms of Levin's u-transform of the ex ponential series:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(x^n/n!,n = 0 .. infinity) = 1+x+x^2/2+x^3/3!+` . . . `+x^n/n!+` . . . `;" "6#/-%$SumG6$*&)%\"xG%\"nG\"\"\"-%*factorialG6#F*!\"\"/F*;\" \"!%)infinityG,0F+F+F)F+*&F)\"\"#F6F/F+*&F)\"\"$-F-6#F8F/F+%(~.~.~.~GF +*&)F)F*F+-F-6#F*F/F+F;F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "n := 5:\ni := 'i' : x := 'x':\naa := [seq(x^i/i!,i=0..n-1)];\nbb := levinU(aa):\nfor i \+ from 1 to n do bb[i] := simplify(bb[i]) end do:\nbb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG7'\"\"\"%\"xG,$*$)F'\"\"#F&#F&F+,$*$)F'\"\"$F& #F&\"\"',$*$)F'\"\"%F&#F&\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\" \"\"*&,&!\"\"F$%\"xGF$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$F$,*!#CF$*&\"#=F$F(F$F$*&\"\"'F$F1F $F'*$)F(\"\"$F$F$F'!\"',$*&,(\"#DF$*&\"\"*F$F(F$F$F0F$F$,,\"$+'F$*&\"$ %QF$F(F$F'*&\"$3\"F$F1F$F$*&\"#;F$F=F$F'*$)F(F5F$F$F'\"#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 256 "The following pi cture shows the graph of the exponential function along with the graph of the Taylor polynomial corresponding to the sum of the terms of the original sequence and the graph of the rational function given by the last term of the u-transform." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "f := x -> 24*(25+9*x+x^2)/( 600-384*x+108*x^2-16*x^3+x^4);\nplot([exp(x),f(x),1+x+x^2/2+x^3/6],x=- 3..3,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,$*&,(\"#D\"\"\"9$\"\"**$)F1\"\"#F0F0F0,,\"$+ 'F0F1!$%QF3\"$3\"*$)F1\"\"$F0!#;*$)F1\"\"%F0F0!\"\"\"#CF(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7V7$ $!\"$\"\"!$\"1$R'yOoqy\\!#<7$$!1+++vq@pG!#:$\"1,!*)o\\LVn&F-7$$!1++D^N UbFF1$\"1XlPp$3#ejF-7$$!1++]K3XFEF1$\"14#R.nViA(F-7$$!1++]F)H')\\#F1$ \"1IY*=ja(>#)F-7$$!1++D'3@/P#F1$\"1/Bh6r8W$*F-7$$!1++Dr^b^AF1$\"1_04\" QaB0\"!#;7$$!1++D,kZG@F1$\"1u^1A\\=!>\"FM7$$!1++Dh\")=,?F1$\"1tt(eyXFM7$$!1+++0)H%*\\\"F1$\"1/HHTUdKAFM7$$!1+++v l[p8F1$\"1I?;zXPUDFM7$$!1+++&>iUC\"F1$\"1M)HwIM:)GFM7$$!1++DhkaI6F1$\" 19X)[zn&GKFM7$$!1+++]XF`**FM$\"1!*[tlR-'p$FM7$$!1++++Az2))FM$\"1NLPs(=y()FM7$$!1b+++v` hH!#=$\"1oDEN%G/(**FM7$$\"1++](QIKH\"FM$\"1***G:qd!Q6F17$$\"1****\\7:x WCFM$\"1i+5&[`pF\"F17$$\"1,++vuY)o$FM$\"1%)oYtf1Y9F17$$\"1)******4FL( \\FM$\"1&)>f7&HVk\"F17$$\"1)****\\d6.B'FM$\"1rX*RHrX'=F17$$\"1++](o3lW (FM$\"1%Q9)Qhq0@F17$$\"1*****\\A))oz)FM$\"1^rQ0'\\,T#F17$$\"1+++Ik-,5F 1$\"113!pQt5s#F17$$\"1+++D-eI6F1$\"1z`u4$F17$$\"1++v=_(zC\"F1$\"1y nSEHG$[$F17$$\"1+++b*=jP\"F1$\"1??'Gt'HgRF17$$\"1++v3/3(\\\"F1$\"1=db_ MioWF17$$\"1++vB4JB;F1$\"1'=I/W[)p]F17$$\"1+++DVsYw7#F1$\"1*e*)zGb[R)F17$$\"1++v)Q?QD#F1$\"1JhP-?0C&*F17$$\"1+++5j ypBF1$\"1@wDg1^p5!#97$$\"1++]Ujp-DF1$\"1$=shkQ:A\"Fby7$$\"1+++gEd@EF1$ \"1RVI?Utv8Fby7$$\"1+]PMh%\\o#F1$\"1-\\$G=TdY\"Fby7$$\"1++v3'>$[FF1$\" 1&GC:#ojh:Fby7$$\"1+++5h(*3GF1$\"1*3;$>?Hf;Fby7$$\"1++D6EjpGF1$\"1@&*y uR0j>'>\"FM7$FT$\"1`g:0!H kN\"FM7$FY$\"16U%)[8=Q:FM7$Fhn$\"18:fJ+=^,Y#=N'>FM7$Fbo$ \"1qr7jG#RB#FM7$Fgo$\"1k#Ru#\\EVDFM7$F\\p$\"1[Yd'z,@)GFM7$Fap$\"1.Lqsd #*GKFM7$Ffp$\"1:)H)=^@'p$FM7$F[q$\"1K[_m9qWTFM7$F`q$\"1!*RF*G!QNZFM7$F eq$\"1X*G/DR\"G`FM7$Fjq$\"1P'*[jY@kgFM7$F_r$\"1\\e!*R'RF17 $F[w$\"1I/]A(Q=Z%F17$F`w$\"1zU-%3he2&F17$Few$\"17$pgU*[YdF17$Fjw$\"1! \\T^5Oea'F17$F_x$\"1i?zl0zBuF17$Fdx$\"1%oMfxS$[%)F17$Fix$\"1J\"QB@06h* F17$F^y$\"1l1fW-\"H3\"Fby7$Fdy$\"1B#***eL8V7Fby7$Fiy$\"1X:KcFP39Fby7$F ^z$\"1Suc\"fBi]\"Fby7$Fcz$\"1?\"emh#p6;Fby7$Fhz$\"1(f(H9oX?Fby7$Fg[l$\"1$yM/8R<7#Fby-F\\\\l6&F^ \\lF*F_\\lF*-F$6$7S7$F($!\"#F*7$F/$!1.bmQSx*o\"F17$F5$!19MW'=QfW\"F17$ F:$!1!)[L,@!))>\"F17$F?$!1a&\\&RaTp(*FM7$FD$!1Gf))HLJ3yFM7$FI$!1=X*Qv: =>'FM7$FO$!1S7y*piTq%FM7$FT$!17ZE2?AXLFM7$FY$!1a^)*et4_@FM7$Fhn$!1/a-L -Zr5FM7$F]o$!1Bq?uWD`AF-7$Fbo$\"1%GJD1HcG'F-7$Fgo$\"1f]jV(H=S\"FM7$F\\ p$\"1S'ev\"Ht(3#FM7$Fap$\"1Mh1!*H*on#FM7$Ffp$\"1UpNGipcLFM7$F[q$\"1p], $\\hA$RFM7$F`q$\"1`I*y!=\\AYFM7$Feq$\"1L!pjf1,F&FM7$Fjq$\"18%[\"yh`SgF M7$F_r$\"1r5+1Tk^oFM7$Fdr$\"1$RH\")o!4*z(FM7$Fir$\"1^/`E12y()FM7$F^s$ \"1O0BN%G/(**FM7$Fds$\"1Sh+Rd/Q6F17$Fis$\"1&)4H_qzw7F17$F^t$\"1mkR\"=M _W\"F17$Fct$\"1!eZ(>V]T;F17$Fht$\"1$H3Q%=Ud=F17$F]u$\"1FO>)3A24#F17$Fb u$\"1&>bgDt+Q#F17$Fgu$\"1XaS&zL#pEF17$F\\v$\"1f6\"e^R0,$F17$Fav$\"1dA* f#yj]LF17$Ffv$\"1Xx9Q@'zv$F17$F[w$\"15teKr#p<%F17$F`w$\"1!*>#4e@Ql%F17 $Few$\"17\"H5$*p/;&F17$Fjw$\"18wPh)Rdt&F17$F_x$\"1LN1KM:NjF17$Fdx$\"1u ZcV\\?'*pF17$Fix$\"1Z3B\\yz,xF17$F^y$\"1J2'RI0eR)F17$Fdy$\"1*Qc>mUqC*F 17$Fiy$\"1emn&)\\215Fby7$Fcz$\"161/oXZ)4\"Fby7$F][l$\"1ySyA*\\D>\"Fby7 $Fg[l$\"#8F*-F\\\\l6&F^\\lF_\\lF_\\lF*-%+AXESLABELSG6$Q\"x6\"%!G-%*THI CKNESSG6#\"\"#-%%VIEWG6$;F(Fg[l%(DEFAULTG" 1 2 0 1 10 2 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. \+ infinity);" "6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "We construct the first \+ few terms of Levin's u-transform of the p-series:" }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 0 .. infinity) = 1+1/ 4+1/9+1/16+` . . . `+1/n^2+` . . . `" "6#/-%$SumG6$*&\"\"\"F(*$%\"nG\" \"#!\"\"/F*;\"\"!%)infinityG,0F(F(*&F(F(\"\"%F,F(*&F(F(\"\"*F,F(*&F(F( \"#;F,F(%(~.~.~.~GF(*&F(F(*$F*F+F,F(F8F(" }{XPPEDIT 18 0 "`` = Pi^2/6; " "6#/%!G*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "n := 10: i : = 'i':\naa := [seq(1/i^2,i=1..n)];\nbb := levinU(aa,info=true);\nevalf (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG7,\"\"\"#F&\"\"%#F&\"\"* #F&\"#;#F&\"#D#F&\"#O#F&\"#\\#F&\"#k#F&\"#\")#F&\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>sequence~ of~partial~sums~...~G7,\"\"\"#\"\"&\"\"%#\"#\\\"#O#\"$0#\"$W\"#\"%p_\" %+O#\"%p`F1#\"'\"om#\"'+k<#\"(\\x2\"\"'+cq#\"(T\"y(*\"(+/N'#\"(H$o>\"( !3q7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~sequence~...~G7,\"\"\"#F%\"\"##F%\"\"$#F%\" \"%#F%\"\"&#F%\"\"'#F%\"\"(#F%\"\")#F%\"\"*#F%\"#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;INITIAL~QUOT IENT~SEQUENCE:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8numerator~sequenc e~...~G7,\"\"\"#\"\"&\"\"##\"#\\\"#7#\"$0#\"#O#\"%p_\"$?(#\"%p`\"$+'# \"'\"om#\"&+_##\"(\\x2\"\"&+#))#\"(T\"y(*\"'+cq#\"(H$o>\"'3q7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%:denominator~sequence~...~G7,\"\"\"\" \"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5DIVIDED~DIFFER ENCES:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8numerator~sequence~...~G7 ,\"\"\"#!\"$\"\"%#\"#8\"#O#!$b$\"%/B#\"$z$\"%+g#!'toB\"(+7L*#\"&z`#\"( ]5_##!)L\\$4'\",+s19a\"#\"*^P(Ri\"-o6b$)\\S#!-B!>\"G\"e(\"1+++++3q7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%:denominator~sequence~...~G7,\"\"\"# !\"\"\"\"##F(\"\"*#!\"$\"#K#\"#C\"$D'#!\"&\"$C$#\"$?(\"'\\w6#!$:$\"'s5 8#\"%![%\"(pHy%#!$n&\"(+Dc\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG7,\"\"\"#\"\"$\"\"##\"#8\"\")# \"$b$\"$;##\"%&*=\"%_6#\"'toB\"'+S9#\"'`w<\"'+!3\"#\")L\\$4'\")+S/P#\" *^P(Ri\"*g0Lz$#\"-B!>\"G\"e(\"-+/j')3Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,$\"\"\"\"\"!$\"+++++:!\"*$\"++++D;F)$\"+>&=Nk\"F)$\"+y_'\\k\"F )$\"+*Q^\\k\"F)$\"+&=N\\k\"F)$\"+VR$\\k\"F)$\"+TS$\\k\"F)$\"+mS$\\k\"F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi^2/6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +oS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Sum (i*x^(i-1),i = 1 .. infinity)=1/(x-1)^2" "6#/-%$SumG6$*&%\"iG\"\"\")% \"xG,&F(F)F)!\"\"F)/F(;F)%)infinityG*&F)F)*$,&F+F)F)F-\"\"#F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "We construct the first few ter ms of Levin's u-transform the series:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(i*x^(i-1),i = 1 .. infinity) = 1+2*x+3*x^2+4* x^3+` . . . `+i*x^(i-1)+` . . . `;" "6#/-%$SumG6$*&%\"iG\"\"\")%\"xG,& F(F)F)!\"\"F)/F(;F)%)infinityG,0F)F)*&\"\"#F)F+F)F)*&\"\"$F)*$F+F3F)F) *&\"\"%F)*$F+F5F)F)%(~.~.~.~GF)*&F(F))F+,&F(F)F)F-F)F)F:F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "n := 5:\ni := 'i':x := 'x':\naa := [seq(i*x^(i-1),i =1..n)];\nbb := levinU(aa):\nfor i from 1 to n do bb[i] := simplify(bb [i]) end do:\nbb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG7'\"\"\",$% \"xG\"\"#,$*$)F(F)F&\"\"$,$*$)F(F-F&\"\"%,$*$)F(F1F&\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"\"*&,&!\"\"F$*&\"\"#F$%\"xGF$F$F$,&F'F$*& \"\"%F$F*F$F$F',$*&,&F'F$F*F$F$,(F$F$*&\"\"$F$F*F$F'*&F3F$)F*F)F$F$F'F '*&F$F$,(*$F5F$F$*&F)F$F*F$F'F$F$F'F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 132 "From the 4 th term onwards the terms of the u-transform become constant and equal to the exact expression \+ for the sum of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "i := 'i': x := 'x':\nSum(i*x^(i-1), i=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6 $*&%\"iG\"\"\")%\"xG,&F'F(!\"\"F(F(/F';F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$),&!\"\"F$%\"xGF$\"\"#F$F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Limit(arctan(n),n = infinity); " "6#-%&LimitG6$-%'arctanG6#%\"nG/F)%)infinityG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 69 "We construct the first few terms of Levin 's u-transform the sequence " }{XPPEDIT 18 0 "a[n]=arctan(n" "6#/&%\"a G6#%\"nG-%'arctanG6#F'" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "n := 8: i := 'i':\naa : = [seq(evalf(arctan(i)),i=1..n)];\nbb := levinU(aa,mode=sequence);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#aaG7*$\"+N;)R&y!#5$\"+=([r5\"!\"*$ \"+sd/\\7F+$\"+kw\"eK\"F+$\"+n2St8F+$\"+\\wk09F+$\"+s#**)G9F+$\"+K8WY9 F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bbG7*$\"+K;)R&y!#5$\"+uGGmD! \"*$\"+b]k(Q\"F+$\"+a2Oq:F+$\"+Si+w:F+$\"+[8(*p:F+$\"+%)Hxq:F+$\"+12#3 d\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The limit of the sequence is " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\" \"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fjzq:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {XPPEDIT 18 0 "Sum(ln(n)/(n^2),n = 2 .. infinity);" "6#-%$SumG6$*&-% #lnG6#%\"nG\"\"\"*$F*\"\"#!\"\"/F*;F-%)infinityG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 70 "We construct the first few terms of Levin 's u-transform of the series:" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(ln(n)/(n^2),n = 2 .. infinity) = ln(2)/4+ln(3)/9+ln (4)/16+` . . . `+ln(n)/(n^2)+` . . . `;" "6#/-%$SumG6$*&-%#lnG6#%\"nG \"\"\"*$F+\"\"#!\"\"/F+;F.%)infinityG,.*&-F)6#F.F,\"\"%F/F,*&-F)6#\"\" $F,\"\"*F/F,*&-F)6#F7F,\"#;F/F,%(~.~.~.~GF,*&-F)6#F+F,*$F+F.F/F,FAF," }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "n := 15: i := 'i':\naa := [seq(ln(i)/i^2,i=2. .n)]:\nbb := levinU(aa):\nevalf(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #70$\"+_z'Gt\"!#5$!+:\\t_7F&$\"+0dZQhF&$\"+8Dd!e*F&$\"+E*Qe`*F&$\"+O\\ .#\\*F&$\"+BX7g%*F&$\"+ejZQ%*F&$\"+9ItB%*F&$\"+Rj38%*F&$\"+2w+1%*F&$\" +g)3xR*F&$\"+g')4+%*F&$\"+Q,%GP*F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "n := 15: i := 'i':\naa := [ seq(ln(i)/i^2,i=2..n)]:\nbb := levinU(aa):\nevalf(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#70$\"+_z'Gt\"!#5$!+9\\t_7F&$\"+zcZQhF&$\"+ZDd!e*F& $\"+q#Re`*F&$\"+Hg.#\\*F&$\"+^&=,Y*F&$\"+/xYQ%*F&$\"+!zVPU*F&$\"+Rj38% *F&$\"+%RCeS*F&$\"+e<()*R*F&$\"+kY<*R*F&$\"+\")e:%Q*F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The terms of the ser ies do no tend to 0 fast enough for the transformed sequence to conver ge." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "i := 'i':\nSum(ln(i)/i^2,i=1..infinity);\nvalue(%);\n evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%#lnG6#%\"iG \"\"\"*$)F*\"\"#F+!\"\"/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%%ZetaG6$\"\"\"\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+VD[v$*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 ";\023" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "A procedure for performing summation using Levin's u-tran sform: " }{TEXT 0 8 "levinsum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "levinsum" } {TEXT -1 43 " is based on the Maple built-in procedure " }{TEXT 0 12 "`evalf/Sum1`" }{TEXT -1 19 ", which is used by " }{TEXT 0 9 "evalf/Su m" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "levinsum: usage" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 12 " levinsum( " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 2 ", " }{TEXT 261 1 "n" } {TEXT -1 14 "=m..infinity) " }}{PARA 0 "" 0 "" {TEXT -1 17 " levinval ue(Sum(" }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 2 ", " } {TEXT 269 1 "n" }{TEXT -1 14 "=m..infinity))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 " " {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 3 " " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT 262 4 " - " }{TEXT -1 51 " an expr ession involving a single variable, say n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 23 7 "m - " } {TEXT -1 29 "the initial index of the sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "levinsum" }{TEXT -1 56 " \+ attempts to evaluate the sum of an alternating series " }{XPPEDIT 18 0 "Sum(a[n],n = m .. infinity);" "6#-%$SumG6$&%\"aG6#%\"nG/F);%\"mG%)i nfinityG" }{TEXT -1 35 " by applying Levin's u transform.\n" }}{PARA 0 "" 0 "" {TEXT 260 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 94 "numterms=n\nThis option can be used to control the number of terms of the series to be handled." }}{PARA 0 "" 0 "" {TEXT -1 35 "The default is \"numterms=Digits*2\"." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 153 "info=true/false,0, 1, 2 or 3\nThis o ption allows the progress of the procedure to be monitored at differen t levels.\n\"info=true\" is equivalent to \"info=3\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 15 "How to activate" }{TEXT 256 2 ":\n" }{TEXT -1 155 "To ma ke the procedure active, open the subsection, place the cursor anywher e after the prompt [ > and press [Enter].\nYou can then close up the \+ subsection." }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 16 ": The \+ procedure " }{TEXT 0 8 "levinsum" }{TEXT -1 54 " may be invoked by mea ns of the pair of commands: \n " }{TEXT 0 22 "Sum(??,n=m..infinity); " }{TEXT -1 4 "\n " }{TEXT 0 14 "levinvalue(%);" }{TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "levinsum: implemetation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6447 " levinvalue := proc(ff)\n local ok,an,rng;\n ok := false;\n if ty pe(ff,'function') and op(0,ff)='Sum' and nops(ff)=2 then\n an := \+ op(1,ff);\n rng := op(2,ff);\n if type(an,algebraic) and typ e(rng,name=integer..infinity) then\n ok := true;\n end if ;\n end if; \n if ok then\n levinsum(an,rng,args[2..nargs]) \n else\n error \"the 1st argument must have the form: Sum(a(n) ,n=m..infinity)\"\n end if;\nend proc: # levinvalue\n\nlevinsum := p roc(ex::algebraic,eq::name=integer..infinity)\n\nlocal Options,prntflg ,x,n,a,d,eps,i,j,k,r,L,r0,r1,r2,\n s0,s1,s2,n0,levinu;\n\n# procedur e for constructing the u-transform\nlevinu := proc(aa)\n local a,n,n um,den,k,sk,ak,j,f,f1,f2,r,kj,kjp;\n a := aa;\n n := op(2, op(2, o p(a)));\n num := array(0..n);\n den := array(0..n);\n r := array (0..n);\n sk := a[0];\n r[0] := sk;\n den[0] := 1/sk;\n num[0] := 1;\n for k to n do\n ak := a[k];\n sk := sk + ak;\n \+ den[k] := 1/((k + 1)^2*ak);\n num[k] := sk*den[k];\n f1 : = k/(k + 1);\n f2 := 1/k;\n for j to k do\n kj := k \+ - j;\n kjp := kj + 1;\n f2 := f2*f1;\n f := kj p*f2;\n num[kj] := num[kjp] - f*num[kj];\n den[kj] := \+ den[kjp] - f*den[kj]\n end do;\n if den[0]=0 then r[k] := in finity else r[k] := num[0]/den[0] end if\n end do;\n eval(r)\nend \+ proc: # of levinu\n\n # start of main procedure levinsum \n x := op(1,eq);\n if not type(indets(ex,name) minus \{x\},set(realcons)) \+ then\n error \"the 1st argument, %1, is invalid .. it should be a n expression which depends only on the variable %2\",ex,x;\n end if; \n L := op(1,op(2,eq));\n \n # Get the options \"numterms\" and \+ \"info\".\n # Set the default values to start with.\n n0 := Digits *2;\n prntflg := 0;\n if nargs > 2 then\n Options:=[args[3..n args]];\n if not type(Options,list(equation)) then\n erro r \"each optional argument must be an equation\"\n end if;\n \+ if hasoption(Options,'numterms','n0','Options') then\n if not (type(n0,integer) and n0 > 1) then\n error \"\\\"numterms \\\" must be an integer greater than 1\"\n end if;\n n 0 := n0 - 1;\n end if;\n if hasoption(Options,'info','prntfl g','Options') then\n if not member(prntflg,\{true,false,0,1,2, 3\}) then\n error \"info must be false=0,1,2 or true=3\"\n \+ end if;\n if prntflg=false then prntflg := 0\n \+ elif prntflg=true then prntflg := 3 end if; \n end if;\n if nops(Options)>0 then\n error \"%1 is not a valid option for % 2\",op(1,Options),procname;\n end if;\n end if;\n\n # Increas e precision for the computation\n d := Digits;\n eps := Float(1,-d -1);\n Digits := 2*d + length(d);\n a := array(0..n0);\n j := -1 ;\n\n # Set up an array of non-zero terms of the series\n for i fr om 0 to 4*n0 while j < n0 do\n r := traperror(evalf(eval(subs(x=L +i,ex))));\n if r = lasterror or not type(r, 'complex'('numeric') ) then\n error \"a non-numeric value occurred\"\n end if; \n if r <> 0 then j := j + 1; a[j] := r end if;\n if j = -1 \+ and n0 < i then return 0 end if\n end do;\n\n # if array is not f illed, shorten it to filled length\n if j <> n0 then a := array(0 .. j, op(3,eval(a))) end if;\n if prntflg > 2 then\n printf(`Term s of original series:`);\n print(convert(a,list));\n print(` `);\n end if;\n\n # Try just summing terms to see if the partial s ums converge\n if prntflg > 0 then\n print(`Trying ordinary sum mation first.`);\n end if;\n s1 := a[0];\n if prntflg > 2 then p rint(`partial sum `||1||` -> `,s1) end if;\n s2 := s1 + a[1];\n \+ if prntflg > 2 then print(`partial sum `||2||` -> `,s2) end if;\n \+ for i to j - 1 do\n s0 := s1;\n s1 := s2;\n s2 := s1 \+ + a[i + 1];\n if prntflg > 2 then\n k := i+2;\n p rint(`partial sum `||k||` -> `,s2)\n end if;\n if s1 = 0 \+ then\n if abs(s0) < eps and abs(s2) < eps then\n if prntflg > 0 then\n k := i+3;\n prin t(`The sum appears to have converged after adding `||k||` terms.`);\n \+ end if;\n return 0\n end if\n \+ else\n if abs(1 - s0/s1) < eps and abs(1 - s2/s1) < e ps then\n if prntflg > 0 then\n k := i+ 3;\n print(`The sum appears to have converged after a dding `||k||` terms.`);\n end if;\n return evalf(s1,d)\n end if\n end if\n end do;\n\n # \+ try Levin's u-transform\n if prntflg > 0 then\n k := i+1;\n \+ print(`Summed `||k||` terms to `,evalf(s2,d),` without convergence.` );\n print(`Constructing Levin's u-transform ...`);\n end if;\n r := levinu(a);\n if prntflg > 2 then\n print(``);\n pr intf(`Terms of transformed series:`);\n print(convert(r,list));\n print(``);\n end if;\n if prntflg > 0 then\n print(`... and checking for convergence.`);\n end if;\n r1 := r[0];\n if p rntflg > 1 then print(`term `||1||` -> `,r1) end if;\n r2 := r[1] ;\n if prntflg > 1 then print(`term `||2||` -> `,r2) end if;\n \+ for i to j - 1 do\n r0 := r1;\n r1 := r2;\n r2 := r[i + 1];\n if prntflg > 1 then\n k := i+2;\n print(`t erm `||k||` -> `,r2)\n end if;\n if type(r0, 'complex'('n umeric')) and type(r1, 'complex'('numeric')) and\n type(r2, 'comp lex'('numeric')) then\n if r1 = 0 then\n if abs(r0) < eps and abs(r2) < eps then\n if prntflg > 0 then\n \+ k := i+3;\n print(`Levin's u-transform appears to have converged after `||k||` terms.`);\n \+ print(``);\n end if;\n return 0;\n \+ end if;\n else\n if abs(1 - r0/r1) < eps and ab s(1 - r2/r0) < eps then\n if prntflg > 0 then\n \+ k := i+2;\n print(`Levin's u-transform appea rs to have converged after `||k||` terms.`);\n print( ``);\n end if;\n return evalf(r1,d)\n \+ end if;\n end if;\n else\n if r0 = infinit y and r1 = infinity and r2 = infinity then\n return infinit y\n elif r0 = -infinity and r1 = -infinity and r2 = -infinity \+ then\n return -infinity\n end if;\n end if;\n \+ end do;\n k := i+1;\n error \"Levin's u-transform with %1 terms \+ has not converged.\", k\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 8 "levinsu m" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 83 "Levin's u transform method method works quite well w ith alternating series such as." }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum((-1)^(n-1)/sqrt(n),n=1..infinity)" "6#-%$SumG6$*&), $\"\"\"!\"\",&%\"nGF)F)F*F)-%%sqrtG6#F,F*/F,;F)%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "n := 'n':\nlevinsum((-1)^(n-1)/sqrt(n),n=1..infinity, info=true);" }}{PARA 6 "" 1 "" {TEXT -1 25 "Terms of original series: " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#77$\"\"\"\"\"!$!75SCva'=\"y1rq!#A$ \"7*3XwD'*=p-Nx&F)$!7++++++++++]F)$\"7=GRz&**\\&f8sWF)$!7iO;I'QY!H[#3% F)$\"7Z@FsA4IZkzPF)$!7/?iPF$f!R`NNF)$\"7LLLLLLLLLLLF)$!7**>Lz$o,mxA;$F )$\"7okAOwxXM6:IF)$!7YD#)G\"[fM^n)GF)$\"7\"45c9E6)4]tFF)$!7Yo%QCC\">Ch sEF)$\"7'yc7hru*)))>e#F)$!7++++++++++DF)$\"7*=N(HLO]iNDCF)$!7qYTe^&RgA qN#F)$\"7s!fwh0(Qt:%H#F)$!74kp*y*\\xz1OAF)$\"7hE\"Q#*fB!*y@=#F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%ATrying~ordinary~summation~first.G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%5partial~sum~1~~->~~~G$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~2~~->~~~G$\"7!*fvCX8)=K*GH!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~3~~->~~~G$\"7z5S#yI+)[V-()!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~4~~->~~~G$\"7z5S#yI+)[V-P!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~5~~->~~~G$\"7(*Qzh..N3 du\")!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~6~~->~~~G$\" 7N-jJ~~~G$\"7#Q-R+%[gEtry!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~ sum~8~~->~~~G$\"7y.Gm7ba()>OV!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5 partial~sum~9~~->~~~G$\"76Ph*f%)y3K&pw!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~10~~->~~~G$\"77~~~G$\"7!=3l&Q\\tyOAv!#A" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~12~~->~~~G$\"7McoFdaFlh NY!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~13~~->~~~G$\"7D dHt=n3v64u!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~14~~->~ ~~G$\"7z)[%Hwa*30lt%!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~s um~15~~->~~~G$\"7lcqS#>q)R\\=t!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 6partial~sum~16~~->~~~G$\"7lcqS#>q)R\\=[!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~17~~->~~~G$\"7a3WqDQP-&QC(!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~18~~->~~~G$\"7%=E?TFMjFo)[!#A " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~19~~->~~~G$\"7c_oHI8 s\\)4=(!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~20~~->~~~G $\"7Z)))*RKj%*p\"\\%\\!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial ~sum~21~~->~~~G$\"73:!Q;$*p*e4Fr!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%4Summed~21~terms~to~G$\"+(*e4Fr!#5%6~without~convergence.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%EConstructing~Levin's~u-transform~...G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1 28 "Te rms of transformed series:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#77$\"\" \"\"\"!$\"73SCva'=\"y1rq!#A$\"7d)*R'HY]A/!**fF)$\"7z&3mh=[=>50'F)$\"7I Hxg*G^eu*[gF)$\"7zr#eje3y\")*[gF)$\"7y0?\"ogat')*[gF)$\"7%HC<*oxIk)*[g F)$\"7w!4b*y[Lk)*[gF)$\"7tF+U*pUV')*[gF)$\"7u~~~G$\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~2~~->~~~G$\"73SCva'=\"y1rq!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~3~~->~~~G$\"7d)*R'HY]A/!**f!# A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~4~~->~~~G$\"7z&3mh=[=>50'! #A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~5~~->~~~G$\"7IHxg*G^eu*[g !#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~6~~->~~~G$\"7zr#eje3y\") *[g!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~7~~->~~~G$\"7y0?\"oga t')*[g!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~8~~->~~~G$\"7%HC<* oxIk)*[g!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~9~~->~~~G$\"7w!4 b*y[Lk)*[g!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~10~~->~~~G$\"7 tF+U*pUV')*[g!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~11~~->~~~G$ \"7u~~~ G$\"7ft-'=;UV')*[g!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnLevin's~u- transform~appears~to~have~converged~after~12~terms.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mk)*[g!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "This result agrees with the result obtained by using Euler's transformatio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eulersum((-1)^(n-1)/sqrt(n),n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Mk)*[g!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 122 "Levin 's u transform method method copes well with series of positive terms \+ for which the terms go to 0 reasonably rapidly." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1 .. infinity);" "6#-%$ SumG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "n := 'n':\nlevinsum(1/n^2,n=1..infinity,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 25 "Terms of original series:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#77$\"\"\"\"\"!$\"7++++++++++D!#A$\"766666666666F)$\"7++ +++++++]i!#B$\"7++++++++++SF.$\"7yxxxxxxxxxFF.$\"7)*[C71`Ej\"3/#F.$\"7 ++++++++]i:F.$\"7N7!zcM7!zcM7F.$\"7++++++++++5F.$\"7,>Pbt\"*4GYk#)!#C$ \"7WWWWWWWWWWpF=$\"7cuY4OJj(fr\"fF=$\"7XAhIlK;3/-^F=$\"7WWWWWWWWWWWF=$ \"7++++++++D1RF=$\"7W[SZnX7w?gMF=$\"7'3`(>k3`(>k3$F=$\"7WAzuI\\-J3qFF= $F(F=$\"7`lrC^9'ptvE#F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%ATrying~ordinary~summation~first.G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~1~~->~~~G$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~2~~->~~~G$\"7+++++++++ ]7!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~3~~->~~~G$\"766 6666666h8!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~4~~->~~~ G$\"766666666hB9!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~5 ~~->~~~G$\"766666666hj9!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partia l~sum~6~~->~~~G$\"7*)))))))))))))))Q\"\\\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~7~~->~~~G$\"7Q8,&>a@0(z6:!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~8~~->~~~G$\"7Q8,&>a@0Au_\"!#@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%5partial~sum~9~~->~~~G$\"7].pSl;JxwR: !#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~10~~->~~~G$\"7].p Sl;Jxw\\:!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~11~~->~~ ~G$\"7Ad/ek(R>K!e:!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum ~12~~->~~~G$\"7m,\\-4UQm(\\c\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 6partial~sum~13~~->~~~G$\"7L'*4;U=)z$*3d\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~14~~->~~~G$\"7X-jU0+Re*fd\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~15~~->~~~G$\"7*ouq)\\W$GS/e\"! #@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~16~~->~~~G$\"7*ouq )\\WLlM%e\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~17~~-> ~~~G$\"7%4AQWd5u1ye\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~ sum~18~~->~~~G$\"7Z=CI0\"3;$*3f\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%6partial~sum~19~~->~~~G$\"7RmJBI\"RCjOf\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~20~~->~~~G$\"7RmJBI\"RCjhf\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6partial~sum~21~~->~~~G$\"7c8Wo\"4w\"3V)f\" !#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%4Summed~21~terms~to~G$\"+=3V)f \"!\"*%6~without~convergence.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ECo nstructing~Levin's~u-transform~...G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 6 "" 1 "" {TEXT -1 28 "Terms of transformed series:" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#77$\"\"\"\"\"!$\"7++++++++++:!#@$\"7++ +++++++D;F)$\"7A&=&=&=&=&=Nk\"F)$\"7nxxxxxx_'\\k\"F)$\"72*))))))))))Q^ \\k\"F)$\"7=_=&=&=&=N\\k\"F)$\"7FeHP'=M%R$\\k\"F)$\"7G.L\"zp6/M\\k\"F) $\"7U!))>aZi1M\\k\"F)$\"74\\Xh([r1M\\k\"F)$\"7D1#Ra$)o1M\\k\"F)$\"7=!* )o[o1M\\k\"F)$\"7gwW^$[o1M\\k\"F)$\"7kU*3$y%o1M\\k\"F)$\"7K A?$H\\o1M\\k\"F)$\"7pdKGe%o1M\\k\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B...~and~checking~for~conver gence.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~1~~->~~~G$\"\"\"\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~2~~->~~~G$\"7++++++++++:!# @" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~3~~->~~~G$\"7+++++++++D;!# @" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~4~~->~~~G$\"7A&=&=&=&=&=Nk \"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~5~~->~~~G$\"7nxxxxxx_' \\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~6~~->~~~G$\"72*)))) ))))))Q^\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~7~~->~~~G$ \"7=_=&=&=&=N\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~8~~->~ ~~G$\"7FeHP'=M%R$\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~9~ ~->~~~G$\"7G.L\"zp6/M\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/ter m~10~~->~~~G$\"7U!))>aZi1M\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %/term~11~~->~~~G$\"74\\Xh([r1M\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~12~~->~~~G$\"7D1#Ra$)o1M\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~13~~->~~~G$\"7=!*)o~~~G$\"7rp?Iw%o1M\\k\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~15~~->~~~G$\"7C/*y=[o1M\\k\"!#@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%hnLevin's~u-transform~appears~to~have ~converged~after~15~terms.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Sum(1/n^2,n= 1..infinity);\nvalue(%);\nevalf(evalf(%,14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*$)%\"nG\"\"#F'!\"\"/F*;F'%)infinity G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"'!\"\"%#PiG\"\"#\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nS$\\k\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 60 "Increasing the number of terms helps in some circumstances. " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(13^n/n!,n = 1 .. in finity);" "6#-%$SumG6$*&)\"#8%\"nG\"\"\"-%*factorialG6#F)!\"\"/F);F*%) infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "levinsum(13^n/n!,n=1..infinity);" } }{PARA 8 "" 1 "" {TEXT -1 74 "Error, (in levinsum) Levin's u-transform with 21 terms has not converged.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "levinsum(13^n/n!,n=1..infi nity,numterms=22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?R7CW!\"%" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 0 9 "evalf/Sum" }{TEXT -1 83 " gives no result, because Maple t ries to use Levin's u transformation, which fails." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(Sum(13 ^n/(n!),n=1..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$* &)\"#8%\"nG\"\"\"-%*factorialG6#F)!\"\"/F);F*%)infinityG" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "However . . ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum(13^n/n!,n=1..infinity);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)\"#8%\"nG\"\"\"-%*factorialG6#F)!\"\"/ F);F*%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"#8\" \"\"F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?R7CW!\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(ln(n)/(n^2),n = 2 .. infinity );" "6#-%$SumG6$*&-%#lnG6#%\"nG\"\"\"*$F*\"\"#!\"\"/F*;F-%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "In this example the u transform goes in the right directi on, but then starts to go haywire." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "levinsum(ln(n)/n^2,n=2..infi nity,info=2,numterms=50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ATrying~ ordinary~summation~first.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%4Summed ~50~terms~to~G$\"+!4tfT)!#5%6~without~convergence.G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%EConstructing~Levin's~u-transform~...G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%B...~and~checking~for~convergence.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~1~~->~~~G$\"7VNFj)*R^z'Gt\"!#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~2~~->~~~G$!74oNlR`:\\t_7!#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~3~~->~~~G$\"7<*z+,Pdmv%Qh!#A" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~4~~->~~~G$\"7T!HO(=/EEd!e*!#A " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~5~~->~~~G$\"7\"))>$*H))G*)Q e`*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~6~~->~~~G$\"7$>zL]hUv N?\\*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~7~~->~~~G$\"78r**4% R=j@,Y*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~8~~->~~~G$\"7Z==I !>[&[ZQ%*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.term~9~~->~~~G$\"7f: -(**=#>FtB%*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~10~~->~~~G$ \"7#e*fCbs8TT8%*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~11~~->~~ ~G$\"74^P\"=c=KpfS*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~12~~- >~~~G$\"7\\tZx!*ya%[/S*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~1 3~~->~~~G$\"7!>\"[PL(fdbiR*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/te rm~14~~->~~~G$\"7XsNLeg5\\+$R*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% /term~15~~->~~~G$\"7#Q`TNxD9R/R*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%/term~16~~->~~~G$\"78'RW<,s&>Q)Q*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~17~~->~~~G$\"7x\"QH>Z&3'4nQ*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~18~~->~~~G$\"7Ze]./$oOL`Q*!#A" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%/term~19~~->~~~G$\"77Uf:pT!Q)=%Q*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~20~~->~~~G$\"7UNBSX&)GkA$Q*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~21~~->~~~G$\"7hk$o!>r)46CQ*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~22~~->~~~G$\"7c9E%edb]9~~~G$\"7P59\\Kt@]6\"Q*!#A " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~24~~->~~~G$\"789/w_>#o&f!Q* !#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~25~~->~~~G$\"7KW1`3_AI9! Q*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~26~~->~~~G$\"7s'Rc)f`( GY(z$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~27~~->~~~G$\"7=t_x Hd^nRz$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~28~~->~~~G$\"7;S 5)RaOK(3z$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~29~~->~~~G$\" 7Aw'RV>x77)y$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~30~~->~~~G $\"7VVug^dWtcy$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~31~~->~~ ~G$\"7\\;%Hdx+DV$y$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~32~~ ->~~~G$\"7+V?V*R5gc\"y$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~ 33~~->~~~G$\"7**yOg\\#>&*ezP*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/ term~34~~->~~~G$\"7M@TQ<~~~G$\"7ZI*=g%Rf!>xP*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~36~~->~~~G$\"7BBg.%oS!*zwP*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~37~~->~~~G$\"7%ot;KJdiW`P*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~38~~->~~~G$\"7V]o1h:qk\\'Q*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~39~~->~~~G$\"7H'4v^]h'[Df$*!#A" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%/term~40~~->~~~G$\"7Yh0C-[m?Pb$*!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~41~~->~~~G$\"7mLEBK'*zm(yg*!#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~42~~->~~~G$\"7m')QUi\\flL$)z!#A " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~43~~->~~~G$\"7+IM\">;Cj$>%G $!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~44~~->~~~G$\"7%G@v'*pGC 6;O\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~45~~->~~~G$\"7THgVJ %y@.**p\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~46~~->~~~G$\"7q >W@eA+;?m=!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~47~~->~~~G$\"7 $p?z;c%RROU>!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~48~~->~~~G$ \"7:+m~~~ G$\"7g-rl\\&fA*z:;!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%/term~50~~-> ~~~G$\"75\"\\r;NDB6eB\"!#@" }}{PARA 8 "" 1 "" {TEXT -1 74 "Error, (in \+ levinsum) Levin's u-transform with 50 terms has not converged.\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "For some \+ reason Maple's " }{TEXT 0 3 "sum" }{TEXT -1 105 " procedure can only o btain this sum if we give the lower index as 1, so that the series has first term 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(ln(n)/n^2,n=1..infinity);\nvalue(%);\nevalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&-%#lnG6#%\"nG\"\"\"F *!\"#/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%%ZetaG6$ \"\"\"\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VD[v$*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 128 "The method works fine with an alternating series wh ose terms have the same magnitude as those of the series in the last e xample." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum((-1)^n* ln(n)/(n^2),n = 2 .. infinity);" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"nGF)-% #lnG6#F+F)*$F+\"\"#F*/F+;F0%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(l evinsum((-1)^n*ln(n)/n^2,n=2..infinity),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"00N;ylJ,\"!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf(evalf(sum((-1)^n*ln(n) /n^2,n=2..infinity),20),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"00N ;ylJ,\"!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6 " }}{PARA 0 "" 0 "" {TEXT -1 17 "We can calculate " }{TEXT 266 18 "Cat alan's constant" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum((-1)^(n-1)/((2*n-1)^2),n = 1 .. infinity);" "6#-%$S umG6$*&),$\"\"\"!\"\",&%\"nGF)F)F*F)*$,&*&\"\"#F)F,F)F)F)F*F0F*/F,;F)% )infinityG" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 54 " 0.91596559 417721901505460351493238411077414937428167 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "See: " }{URLLINK 17 "http://www .mathsoft.com/asolve/constant/catalan/catalan.html" 4 "http://www.math soft.com/asolve/constant/catalan/catalan.html" "" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(levinsum((-1)^(n-1)/(2*n-1)^2,n=1..infinity),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Sn\"Gu$\\Tx5TQK\\^.Y0:!>s<%flf\"*!#]" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Maple \"k nows\" the Catalan constant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(Catalan,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Sn\"Gu$\\Tx5TQK\\^.Y0:!>s<%flf\"*!#]" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Su m((2*n-1)!/(2*n+1)/(2^(2*n-1))/(n-1)!/n!,n = 1 .. infinity)=arcsin(1)- 1" "6#/-%$SumG6$*,-%*factorialG6#,&*&\"\"#\"\"\"%\"nGF.F.F.!\"\"F.,&*& F-F.F/F.F.F.F.F0)F-,&*&F-F.F/F.F.F.F0F0-F)6#,&F/F.F.F0F0-F)6#F/F0/F/;F .%)infinityG,&-%'arcsinG6#F.F.F.F0" }{XPPEDIT 18 0 "``=Pi/2-1" "6#/%!G ,&*&%#PiG\"\"\"\"\"#!\"\"F(F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "Digits := 1 00:\nSum((2*n-1)!/(2*n+1)/(2^(2*n-1))/(n-1)!/n!,n = 1 .. infinity);\nl evinvalue(%,info=1);\nevalf[Digits+5](Pi/2-1):\nevalf(%);\nDigits := 1 0:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*,-%*factorialG6#,&*&\" \"#\"\"\"%\"nGF-F-F-!\"\"F-,&*&F,F-F.F-F-F-F-F/)F,F*F/-F(6#,&F.F-F-F/F /-F(6#F.F//F.;F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ATryin g~ordinary~summation~first.G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%%5Summ ed~201~terms~to~G$\"_q#y@uuy 'p#>4J&!$+\"%6~without~convergence.G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%EConstructing~Levin's~u-transform~...G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B...~and~checking~for~convergence.G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%inLevin's~u-transform~appears~to~have~converged~afte r~115~terms.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qS`e5n7u,9$*\\/J9.#3R:'Hsu[5Hb(o*p%e)4U9vR;p@8B> m*[zEjzq&!$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qS`e5n7u,9$*\\/J 9.#3R:'Hsu[5Hb(o*p%e)4U9vR;p@8B>m*[zEjzq&!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 57 "Find a numerical approximation for th e sum of the series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n^3),n = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"$ !\"\"/F);F'%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by using Levin's u-transform." }}{PARA 0 "" 0 "" {TEXT -1 40 "_______ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "_______________________________ _________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 79 "Use Lev in's u-transformation to find a rational approximation for the functio n " }{XPPEDIT 18 0 "f(x)=sin(x)" "6#/-%\"fG6#%\"xG-%$sinG6#F'" }{TEXT -1 33 " based on the Taylor polynomial " }{XPPEDIT 18 0 "x-x^3/6+x^5/ 120-x^7/5040;" "6#,*%\"xG\"\"\"*&F$\"\"$\"\"'!\"\"F)*&F$\"\"&\"$?\"F)F %*&F$\"\"(\"%S]F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "__ ______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "____________________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 57 "Find a numerical approximation for the sum of the series " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((2*n-1)!/(2^(2*n- 1)*n!*(n-1)!*(2*n+1)),n = 1 .. infinity);" "6#-%$SumG6$*&-%*factorialG 6#,&*&\"\"#\"\"\"%\"nGF-F-F-!\"\"F-**)F,,&*&F,F-F.F-F-F-F/F--F(6#F.F-- F(6#,&F.F-F-F/F-,&*&F,F-F.F-F-F-F-F-F//F.;F-%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "by using Levin's u-transform." }} {PARA 0 "" 0 "" {TEXT -1 40 "________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }