{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 "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 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 262 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasi s" -1 263 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Re d Emphasis" -1 264 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 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 0 } {CSTYLE "Grey Emphasis" -1 270 "Times" 1 12 96 52 84 1 0 1 0 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 "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 "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 23 "Rearrangement of series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "This worksheet uses ideas from Chapter 21 of \"Mathematica in Action\" by Stan Wagon, 2nd edition, S pringer-Verlag." }}{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 270 8 "series.m" }{TEXT -1 38 " contains the code for the procedures " } {TEXT 0 7 "arrange" }{TEXT -1 1 " " }{TEXT -1 24 "used in this workshe et. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple sess ion by a command similar to the one that follows, where the file path \+ gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rea d \"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 48 "Rearrangement of conditionally co nvergent series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 27 "alternating harmonic \+ series" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(k +1)/k,k = 1 .. infinity) = 1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+` . . . `;" " 6#/-%$SumG6$*&),$\"\"\"!\"\",&%\"kGF*F*F*F*F-F+/F-;F*%)infinityG,4F*F* *&F*F*\"\"#F+F+*&F*F*\"\"$F+F**&F*F*\"\"%F+F+*&F*F*\"\"&F+F**&F*F*\"\" 'F+F+*&F*F*\"\"(F+F**&F*F*\"\")F+F+%(~.~.~.~GF*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 21 "converges to the sum " }{XPPEDIT 18 0 "ln (2)" "6#-%#lnG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 132 "This series is not absolutely convergent since the series obtained by taking the absolute values of the terms is the harmonic series" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/k,k = 1 .. infi nity) = 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+` . . . `;" "6#/-%$SumG6$*&\"\" \"F(%\"kG!\"\"/F);F(%)infinityG,4F(F(*&F(F(\"\"#F*F(*&F(F(\"\"$F*F(*&F (F(\"\"%F*F(*&F(F(\"\"&F*F(*&F(F(\"\"'F*F(*&F(F(\"\"(F*F(*&F(F(\"\")F* F(%(~.~.~.~GF(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 27 "which d iverges to infinity." }}{PARA 0 "" 0 "" {TEXT -1 65 "A convergent seri es which does not converge absolutely is called " }{TEXT 262 24 "condi tionally convergent" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 72 "Consider the following rearrangement of \+ the alternating harmonic series:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/2-1/4+1/3-1/6-1/8 +1/5-1/10-1/12+1/7-1/14-1/16+1/9-1/18-1/20+` . . . `;" "6#,B\"\"\"F$*& 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$\"#5F'F'*&F$F$\"#7F'F'*&F$F$\"\"(F'F$*&F$F $\"#9F'F'*&F$F$\"#;F'F'*&F$F$\"\"*F'F$*&F$F$\"#=F'F'*&F$F$\"#?F'F'%(~. ~.~.~GF$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 114 "We can find the symbolic sum of this rearranged s eries by grouping together successive triples of terms thus . . ." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(2*k-1)-1/(4*k- 2)-1/(4*k),k=1..infinity)" "6#-%$SumG6$,(*&\"\"\"F(,&*&\"\"#F(%\"kGF(F (F(!\"\"F-F(*&F(F(,&*&\"\"%F(F,F(F(F+F-F-F-*&F(F(*&F1F(F,F(F-F-/F,;F(% )infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 105 "The follo wing command provides an empirical check that we have the correct form ulas for the denominators." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "seq([1/(2*k-1),-1/(4*k-2),-1/(4*k)] ,k=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7%\"\"\"#!\"\"\"\"##F&\" \"%7%#F$\"\"$#F&\"\"'#F&\"\")7%#F$\"\"&#F&\"#5#F&\"#77%#F$\"\"(#F&\"#9 #F&\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sum(1/(2*k-1)-1/(4*k-2)-1/(4*k),k=1..infinity);\nvalu e(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,(*&\"\"\"F(,&%\"kG \"\"#!\"\"F(F,F(*&F(F(,&F*\"\"%!\"#F(F,F,*&F(F(F*F,#F,F//F*;F(%)infini tyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"##\"\"\"F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "The rearranged series converges to " }{XPPEDIT 18 0 "ln(2)/2" " 6#*&-%#lnG6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 47 ", which is half the sum \+ of the original series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "It turns out that any conditionally convergent serie s can be rearranged so that it " }{TEXT 262 38 "converges to any speci fied real number" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 109 "Inde ed the algorithm to find such a rearrangement of the alternating harmo nic series is very straightforward." }}{PARA 0 "" 0 "" {TEXT -1 40 "In order to converge to a target number " }{TEXT 265 1 "T" }{TEXT -1 75 ", first add just enough successsive positive terms to obtain a partia l sum " }{XPPEDIT 18 0 "S[1]" "6#&%\"SG6#\"\"\"" }{TEXT -1 23 " which \+ is greater than " }{TEXT 266 1 "T" }{TEXT -1 107 ". Then switch to neg ative terms, starting with the first unused negative term, add just en ough of these to " }{XPPEDIT 18 0 "S[1]" "6#&%\"SG6#\"\"\"" }{TEXT -1 23 " to give a partial sum " }{XPPEDIT 18 0 "S[2]" "6#&%\"SG6#\"\"#" } {TEXT -1 20 " which is less than " }{TEXT 267 1 "T" }{TEXT -1 103 ". T hen add positive terms, starting with the first unused positive term, \+ until the sum is greater than " }{TEXT 268 1 "T" }{TEXT -1 108 ". Cont inue to construct groups of terms of alternate common sign by switchin g direction whenever the target " }{TEXT 269 1 "T" }{TEXT -1 16 " is j ust passed." }}{PARA 0 "" 0 "" {TEXT -1 295 "In the case where the ori ginal series is the alternating harmonic series there will always be e nough terms of either sign to reach the target at each stage, because \+ the series consisting of either the positive terms only, and the serie s consisting of the negative terms only, are both divergent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure \+ " }{TEXT 0 7 "arrange" }{TEXT -1 68 " in the next section implements t his algorithm for a general series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 104 "Note that in order to avoid the possibi lity of an infinite loop occurring (which is likely to happen if " } {TEXT 0 7 "arrange" }{TEXT -1 139 " is used with an absolutely converg ent series), a limit is placed on the number of terms of the same sign which can be added at each stage." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "A procedure for rearranging an alternating series: " }{TEXT 0 7 "arrange" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "arrange" }{TEXT -1 124 " uses ideas from a procedure given in \"Mathematica in Action\", by Stan Wagon, 2 nd edition, Springer-Verlag, pages 451,452 . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "arrang e: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 17 " arrange(sm ,t) " }{TEXT 261 1 "\n" }}{PARA 256 "" 0 "" {TEXT -1 12 "Parameters: " }}{PARA 0 "" 0 "" {TEXT 23 8 " sm - " }{TEXT -1 20 " a sum of the form " }{XPPEDIT 18 0 "Sum(a[n],n=m..infinity)" "6# -%$SumG6$&%\"aG6#%\"nG/F);%\"mG%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 23 5 "t - " }{TEXT -1 54 "the target value for the sum of the rearran ged series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedur e " }{TEXT 0 7 "arrange" }{TEXT -1 66 " attempts to construct an arra ngement of the terms of the series " }{XPPEDIT 18 0 "Sum(a[n],n = m .. infinity);" "6#-%$SumG6$&%\"aG6#%\"nG/F);%\"mG%)infinityG" }{TEXT -1 72 " in such a way that the new series converges to the specified tar get t." }}{PARA 0 "" 0 "" {TEXT -1 85 "This is always possible to ache ive when the given series is conditionally convergent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 19 "output=groups/terms" }}{PARA 0 " " 0 "" {TEXT -1 219 "This option can be used to specify whether the ou tput is the list of numbers of terms in the successive groups of posit ive and negative terms or the actual terms of the rearranged series. T he default is \"output=groups\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "numterms=n" }}{PARA 0 "" 0 "" {TEXT -1 80 "This option controls the maximum number of terms of the series to \+ be rearranged." }}{PARA 0 "" 0 "" {TEXT -1 110 "When output=groups the default is \"numterms=Digits*8\" and when output=terms the default is \"numterms=Digits*2\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "groupmax=n" }}{PARA 0 "" 0 "" {TEXT -1 89 "This opti on controls the maximum number of terms in the separate groups of the \+ same sign." }}{PARA 0 "" 0 "" {TEXT -1 44 "The default is \"groupmax=t runc(numterms/2)\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 8 "posmax=n" }}{PARA 0 "" 0 "" {TEXT -1 96 "This option con trols the maximum number terms to check when looking for the next posi tive term ." }}{PARA 0 "" 0 "" {TEXT -1 33 "The default is \"posmax=gr oupmax\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "negmax=n" }}{PARA 0 "" 0 "" {TEXT -1 96 "This option controls the m aximum number terms to check when looking for the next negative term . " }}{PARA 0 "" 0 "" {TEXT -1 33 "The default is \"negmax=groupmax\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "info=t rue/false\nThe option info=true allows the progress of the procedure t o be monitored by printing the partial sums at the sign changes as the y occur." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 17 "How to activate:\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 "arrange: implemetation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4885 "arrange := proc(ff,target::realcons)\n\nlocal Options,n,start,outpt,prntflg,s aveDigits,nmtrms,i,j,k,\n iPos,iNeg,jOld,trms,term,sum,counters,t rgt,mxgrp,\n mxpos,mxneg,eterm,an,eq,m,ok;\n\n ok := false;\n \+ if type(ff,'function') and op(0,ff)='Sum' and nops(ff)=2 then\n \+ an := op(1,ff);\n eq := op(2,ff);\n if type(an,algebraic) an d type(eq,name=anything..infinity) then\n ok := true;\n e nd if;\n end if; \n if not ok then\n error \"the 1st argumen t must have the form: Sum(a(n),n=m..infinity)\"\n end if;\n \n n := op(1,eq);\n if not type(indets(an,name) minus \{n\},set(realcons )) then\n error \"%1 should be an expression which depends only o n the variable %2\",an,n;\n end if;\n m := op(1,op(2,eq));\n if \+ not type(m,integer) then\n error \"initial value in summation ran ge %1 should be an integer\",m;\n end if;\n \n # Get the options .\n # Set the default values to start with.\n outpt := 'groups';\n nmtrms := Digits*8;\n mxgrp := Digits*4;\n mxpos := mxgrp;\n \+ mxneg := mxgrp;\n prntflg := false;\n if nargs>2 then\n Optio ns:=[args[3..nargs]];\n if not type(Options,list(equation)) then \n error \"each optional argument must be an equation\"\n \+ end if;\n if hasoption(Options,'output','outpt','Options') then \n if not (outpt='groups' or outpt='terms') then\n \+ error \"\\\"output\\\" must be 'groups' or 'terms'\"\n end if; \n if outpt='terms' then nmtrms := Digits*2 end if;\n end if;\n if hasoption(Options,'numterms','nmtrms','Options') then\n if not type(nmtrms,posint) then\n error \"\\\"numt erms\\\" must be a positive integer\"\n end if;\n mxgr p := trunc(nmtrms/2);\n mxpos := mxgrp;\n mxneg := mxg rp;\n end if;\n if hasoption(Options,'groupmax','mxgrp','Opt ions') then\n if not type(nmtrms,posint) then\n err or \"\\\"groupmax\\\" must be a positive integer\"\n end if;\n end if;\n if hasoption(Options,'posmax','mxpos','Options') \+ then\n if not type(nmtrms,posint) then\n error \"\\ \"posmax\\\" must be a positive integer\"\n end if;\n end if;\n if hasoption(Options,'negmax','mxneg','Options') then\n \+ if not type(nmtrms,posint) then\n error \"\\\"negmax \\\" must be a positive integer\"\n end if;\n end if;\n \+ if hasoption(Options,'info','prntflg','Options') then\n if prntflg<>true then prntflg := false 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 # Increase precision for the computation\n saveDigits := Digits;\n Digits := Digits + min(trunc(Digits/2),10);\n\n sum := 0;\n j := m;\n iNe g := m;\n iPos := m;\n jOld := m;\n counters := NULL;\n trms : = NULL;\n trgt := evalf(target);\n\n do\n i := 0;\n whil e sum<=trgt and i `,sum)\n end if;\n counters := counte rs,j-jOld;\n jOld := j;\n if j>nmtrms then break end if;\n \+ i := 0;\n while sum >= trgt and i < mxgrp do\n i := i +1;\n term := 1;\n k := 1;\n while term>0 do\n if k<=mxneg then\n term := eval(subs(n=iNeg, an));\n iNeg := iNeg+1;\n k := k+1;\n \+ else\n error \"exceeded maximum number, %1, of ne gative terms\",mxneg;\n end if; \n end do; \+ \n eterm := traperror(evalf(term));\n if eterm = las terror or not type(eterm,numeric) then\n error \"a non-nume ric value occurred\"\n end if;\n sum := sum+eterm;\n \+ trms:= trms,term;\n end do;\n if i=mxgrp then\n \+ error \"reached maximum number, %1, of grouped terms\",mxgrp;\n \+ end if;\n j := j+i;\n if prntflg then\n print(`par tial sum with `||i||` more negative terms -> `,sum)\n end if; \n counters := counters,j-jOld;\n jOld := j;\n if j>nmt rms then break end if;\n end do;\n if outpt='terms' then\n [t rms]\n else\n [counters]\n end if;\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 -1 0 "" }{TEXT 0 7 "arrange" }{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 13 "Consider the \+ " }{TEXT 262 27 "alternating harmonic series" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(k+1)/k,k = 1 .. i nfinity) = 1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+` . . . `;" "6#/-%$SumG6$*&), $\"\"\"!\"\",&%\"kGF*F*F*F*F-F+/F-;F*%)infinityG,4F*F**&F*F*\"\"#F+F+* &F*F*\"\"$F+F**&F*F*\"\"%F+F+*&F*F*\"\"&F+F**&F*F*\"\"'F+F+*&F*F*\"\"( F+F**&F*F*\"\")F+F+%(~.~.~.~GF*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(2);" "6#/%!G-%#lnG6#\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sm := Sum((-1)^(n+1)/n,n=1..infinity);\nvalue(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#smG-%$SumG6$*&)!\"\",&%\"nG\" \"\"F-F-F-F,F*/F,;F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%# lnG6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "The alternating harmonic series can be reconstructed using " } {TEXT 0 7 "arrange" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "Th e output is the list of numbers of terms in the successive groups of p ositive and negative terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "arrange(sm,ln(2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7\\p\"\"\"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$F$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$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "arrange(sm,l n(2),output=terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"#!\"\" \"\"##F$\"\"$#F&\"\"%#F$\"\"&#F&\"\"'#F$\"\"(#F&\"\")#F$\"\"*#F&\"#5#F $\"#6#F&\"#7#F$\"#8#F&\"#9#F$\"#:#F&\"#;#F$\"#<#F&\"#=#F$\"#>#F&\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Consider the rearrangemen t: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/2-1/4+1/3- 1/6-1/8+1/5-1/10-1/12+1/7-1/14-1/16+1/9-1/18-1/20+` . . . `;" "6#,B\" \"\"F$*&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$\"#5F'F'*&F$F$\"#7F'F'*&F$F$\"\"(F' F$*&F$F$\"#9F'F'*&F$F$\"#;F'F'*&F$F$\"\"*F'F$*&F$F$\"#=F'F'*&F$F$\"#?F 'F'%(~.~.~.~GF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "corres ponding to the groupings " }{XPPEDIT 18 0 "1, 2, 1, 2, 1, 2, 1, 2, 1, \+ 2, 1, 2, ` . . . `" "6/\"\"\"\"\"#F#F$F#F$F#F$F#F$F#F$%(~.~.~.~G" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "This series can be summed symbolically by grouping the te rms in triples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "Sum(1/(2*k-1)-1/(4*k-2)-1/(4*k),k=1..infinity) ;\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,(*&\"\"\"F(, &*&\"\"#F(%\"kGF(F(F(!\"\"F-F(*&F(F(,&*&\"\"%F(F,F(F(F+F-F-F-*&F(F(*&F 1F(F,F(F-F-/F,;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&# \"\"\"\"\"#F&-%#lnG6#F'F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The new series can be reconstructed using " } {TEXT 0 7 "arrange" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "arrange(sm,ln(2)/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7X\"\"\"\"\"#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$F%F$F%F$F%F$F%F$F%F$F %F$F%F$F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "arrange(sm,ln(2)/2,output=terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#77\"\"\"#!\"\"\"\"##F&\"\"%#F$\"\"$#F&\"\"'#F&\"\")# F$\"\"&#F&\"#5#F&\"#7#F$\"\"(#F&\"#9#F&\"#;#F$\"\"*#F&\"#=#F&\"#?#F$\" #6#F&\"#A#F&\"#C#F$\"#8#F&\"#E#F&\"#G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Consider the rearrangement" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+1/3-1/2+1/5+1/7-1/4+1/9+1/11-1/6+1/13+1/15-1/8 +1/17+1/19-1/10+` . . . `;" "6#,B\"\"\"F$*&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$ \"#6F'F$*&F$F$\"\"'F'F'*&F$F$\"#8F'F$*&F$F$\"#:F'F$*&F$F$\"\")F'F'*&F$ F$\"#F'F$*&F$F$\"#5F'F'%(~.~.~.~GF$" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 31 "corresponding to the groupings " } {XPPEDIT 18 0 "2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,` . . . `" "61 \"\"#\"\"\"F#F$F#F$F#F$F#F$F#F$F#F$%(~.~.~.~G" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "This seri es can be summed symbolically by grouping the terms in triples." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sum(1/(4*k-3)+1/(4*k-1)-1/(2*k),k=1..infinity);\nvalue(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,(*&\"\"\"F(,&*&\"\"%F(%\"kGF (F(\"\"$!\"\"F.F(*&F(F(,&*&F+F(F,F(F(F(F.F.F(*&F(F(*&\"\"#F(F,F(F.F./F ,;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\"#\"\" \"-%#lnG6#F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The new series can be reconstructed using " }{TEXT 0 7 "a rrange" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "arrange(sm,3*ln(2)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7W\"\"#\"\"\"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$F%F$F%F$F%F$F%F$F%F$F%F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "arrange(sm,3*ln(2)/2,output=terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"#F$\"\"$#!\"\"\"\"##F$\"\"&#F$\"\"(#F(\"\"%#F$ \"\"*#F$\"#6#F(\"\"'#F$\"#8#F$\"#:#F(\"\")#F$\"#<#F$\"#>#F(\"#5#F$\"#@ #F$\"#B#F(\"#7#F$\"#D#F$\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Consider the rearrangement" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/2-1/4-1/6+1/3-1/8-1/10- 1/12+1/5-1/14-1/16-1/18+1/7-1/20-1/22-1/24+` . . . `;" "6#,D\"\"\"F$*& F$F$\"\"#!\"\"F'*&F$F$\"\"%F'F'*&F$F$\"\"'F'F'*&F$F$\"\"$F'F$*&F$F$\" \")F'F'*&F$F$\"#5F'F'*&F$F$\"#7F'F'*&F$F$\"\"&F'F$*&F$F$\"#9F'F'*&F$F$ \"#;F'F'*&F$F$\"#=F'F'*&F$F$\"\"(F'F$*&F$F$\"#?F'F'*&F$F$\"#AF'F'*&F$F $\"#CF'F'%(~.~.~.~GF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 " corresponding to the grouping " }{XPPEDIT 18 0 "1,3,1,3,1,3,1,3,1,3,1 ,3,1,3,1,3,1,3,` . . . `;" "65\"\"\"\"\"$F#F$F#F$F#F$F#F$F#F$F#F$F#F$F #F$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 85 "This series can be summed symbolically by grouping the terms in groups of four terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sum(1/(2*k-1)-1/(6 *k-4)-1/(6*k-2)-1/(6*k),k=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$SumG6$,**&\"\"\"F(,&*&\"\"#F(%\"kGF(F(F(!\"\"F-F(* &F(F(,&*&\"\"'F(F,F(F(\"\"%F-F-F-*&F(F(,&*&F1F(F,F(F(F+F-F-F-*&F(F(*&F 1F(F,F(F-F-/F,;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&# \"\"\"\"\"#F&-%#lnG6#\"\"$F&!\"\"-F)6#F'F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The new series can be reconstru cted using " }{TEXT 0 7 "arrange" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "arrange(sm,l n(2)-1/2*ln(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7J\"\"\"\"\"$F$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 %" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "arrange(sm,ln(2)-1/2*ln(3),output=terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76\"\"\"#!\"\"\"\"##F&\"\"%#F&\"\"'#F$\"\"$# F&\"\")#F&\"#5#F&\"#7#F$\"\"&#F&\"#9#F&\"#;#F&\"#=#F$\"\"(#F&\"#?#F&\" #A#F&\"#C#F$\"\"*#F&\"#E#F&\"#G#F&\"#I" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We do not always obtain an identifia ble pattern of terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "arrange(sm,Pi,numterms=500);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"#w\"\"\"\"$H\"F%\"$K\"F%\"$M\"F%\"$L\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 31 "Consider the alternating series" }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1)^(n+1)/(3*n-2),n = 1 .. infin ity) = 1-1/4+1/7-1/10+1/13-1/16+` . . . `;" "6#/-%$SumG6$*&),$\"\"\"! \"\",&%\"nGF*F*F*F*,&*&\"\"$F*F-F*F*\"\"#F+F+/F-;F*%)infinityG,0F*F**& F*F*\"\"%F+F+*&F*F*\"\"(F+F**&F*F*\"#5F+F+*&F*F*\"#8F+F**&F*F*\"#;F+F+ %(~.~.~.~GF*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Pi*sqrt(3)/9+ln(2)/3;" "6#/%!G,&*(%#PiG\"\"\"-%%sq rtG6#\"\"$F(\"\"*!\"\"F(*&-%#lnG6#\"\"#F(F,F.F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sum((-1)^(n+1)/(3*n-2),n=1..infinity);\nvalue(%);\nevalf(evalf(%,4 5),40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\",&%\"nG\" \"\"F+F+F+,&*&\"\"$F+F*F+F+\"\"#F(F(/F*;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%)LerchPhiG6%!\"\"F&F%F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Ilym26+(fM5PL0@ZE[)[c$)!#S" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "By groupi ng the terms together in pairs we obtain the expression given above fo r the sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 72 "Sum(1/(6*k-5)-1/(6*k-2),k=1..infinity);\nvalue(%); \nevalf(evalf(%,45),40);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$, &*&\"\"\"F(,&*&\"\"'F(%\"kGF(F(\"\"&!\"\"F.F(*&F(F(,&*&F+F(F,F(F(\"\"# F.F.F./F,;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"*! \"\"%#PiG\"\"\"\"\"$#F(\"\"#F(*&#F(F)F(-%#lnG6#F+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Ilym26+(fM5PL0@ZE[)[c$)!#S" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The original series can b e reconstructed using " }{TEXT 0 7 "arrange" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sm 2 := Sum((-1)^(n+1)/(3*n-2),n=1..infinity);\narrange(sm2,Pi*sqrt(3)/9+ ln(2)/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sm2G-%$SumG6$*&)!\"\", &%\"nG\"\"\"F-F-F-,&*&\"\"$F-F,F-F-\"\"#F*F*/F,;F-%)infinityG" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7\\p\"\"\"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$F$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$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "arrange(sm2,Pi*sqrt(3)/9+ln(2)/3,output=terms);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#76\"\"\"#!\"\"\"\"%#F$\"\"(#F&\"#5#F$\"#8#F&\"#;#F$\" #>#F&\"#A#F$\"#D#F&\"#G#F$\"#J#F&\"#M#F$\"#P#F&\"#S#F$\"#V#F&\"#Y#F$\" #\\#F&\"#_#F$\"#b#F&\"#e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "Now consider the rearrangement: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/4-1/10+1/7-1/16-1/22+1/13-1/28 -1/34+1/19-1/40-1/46+` . . . `;" "6#,<\"\"\"F$*&F$F$\"\"%!\"\"F'*&F$F$ \"#5F'F'*&F$F$\"\"(F'F$*&F$F$\"#;F'F'*&F$F$\"#AF'F'*&F$F$\"#8F'F$*&F$F $\"#GF'F'*&F$F$\"#MF'F'*&F$F$\"#>F'F$*&F$F$\"#SF'F'*&F$F$\"#YF'F'%(~.~ .~.~GF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "corresponding \+ to the groupings " }{XPPEDIT 18 0 "1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,` . . . `" "60\"\"\"\"\"#F#F$F#F$F#F$F#F$F#F$F#%(~.~.~.~G" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Sum(1/(6*k-5)-1/(12*k-8)-1/(12*k-2),k=1..infinity);\n value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,(*&\"\"\"F(,&*& \"\"'F(%\"kGF(F(\"\"&!\"\"F.F(*&F(F(,&*&\"#7F(F,F(F(\"\")F.F.F.*&F(F(, &*&F2F(F,F(F(\"\"#F.F.F./F,;F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"'F&-%#lnG6#\"\"#F&F&*(\"\"*!\"\"%#PiGF& \"\"$#F&F+F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The new series can be reconstructed using " }{TEXT 0 7 "a rrange" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "arrange(sm2,Pi*sqrt(3)/9+ln(2)/6); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7X\"\"\"\"\"#F$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$F%F$F%F$F%F$F %F$F%F$F%F$F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "arrange(sm2,Pi*sqrt(3)/9+ln(2)/6,output=terms,nu mterms=15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"\"#!\"\"\"\"%#F& \"#5#F$\"\"(#F&\"#;#F&\"#A#F$\"#8#F&\"#G#F&\"#M#F$\"#>#F&\"#S#F&\"#Y#F $\"#D#F&\"#_#F&\"#e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "sigma (n);" "6#-%&sigmaG6#%\"nG" }{TEXT -1 39 " be the sequence of \"signs\" defined by " }{XPPEDIT 18 0 "sigma(n)=sqrt(2)*sin((2*n+1)*Pi/4)" "6#/ -%&sigmaG6#%\"nG*&-%%sqrtG6#\"\"#\"\"\"-%$sinG6#*(,&*&F,F-F'F-F-F-F-F- %#PiGF-\"\"%!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma := n -> sqrt(2)*sin(( 2*n+1)*Pi/4);\nseq(sigma(n),n=1..20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&-%%sqrtG6#\"\"#\"\"\" -%$sinG6#,$*&#F1\"\"%F1*&,&*&F0F19$F1F1F1F1F1%#PiGF1F1F1F1F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"!\"\"F$F#F#F$F$F#F#F$F$F#F#F$F$F #F#F$F$F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the serie s " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(sqrt(2)*sin ((2*n+1)*Pi/4)/n,n = 1 .. infinity) = 1-1/2-1/3+1/4+1/5-1/6-1/7+1/8+1/ 9-1/10-1/11;" "6#/-%$SumG6$*(-%%sqrtG6#\"\"#\"\"\"-%$sinG6#*(,&*&F+F,% \"nGF,F,F,F,F,%#PiGF,\"\"%!\"\"F,F3F6/F3;F,%)infinityG,8F,F,*&F,F,F+F6 F6*&F,F,\"\"$F6F6*&F,F,F5F6F,*&F,F,\"\"&F6F,*&F,F,\"\"'F6F6*&F,F,\"\"( F6F6*&F,F,\"\")F6F,*&F,F,\"\"*F6F,*&F,F,\"#5F6F6*&F,F,\"#6F6F6" } {TEXT -1 9 " + . . . " }}{PARA 0 "" 0 "" {TEXT -1 37 "We can generate \+ terms of this series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(sigma(n)/n,n=1..15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "61\"\"\"#!\"\"\"\"##F%\"\"$#F#\"\"%#F#\"\"&#F%\"\"'# F%\"\"(#F#\"\")#F#\"\"*#F%\"#5#F%\"#6#F#\"#7#F#\"#8#F%\"#9#F%\"#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The terms after the first can be grouped together in pairs to form the alternat ing series" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1 )^k*(4*k+1)/(2*k*(2*k+1)),k=1..infinity)" "6#-%$SumG6$*(),$\"\"\"!\"\" %\"kGF),&*&\"\"%F)F+F)F)F)F)F)*(\"\"#F)F+F),&*&F0F)F+F)F)F)F)F)F*/F+;F )%)infinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 47 "which con verges by the alternating series test." }}{PARA 0 "" 0 "" {TEXT -1 39 "This means that the original series is " }{TEXT 262 24 "conditionally convergent" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We can now calculate the sum of the original se ries, which Maple cannot do directly." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "1+Sum((-1)^k*(4*k+1)/(2* k*(2*k+1)),k=1..infinity);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$SumG6$,$*,\"\"#!\"\")F+%\"kGF$,&*&\"\"%F$F -F$F$F$F$F$F-F+,&*&F*F$F-F$F$F$F$F+F$/F-;F$%)infinityGF$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&#\"\"&\"\"'F$-%*hypergeomG6%7%F$F$# \"\"$\"\"#7$F/#F'F/!\"\"F$F2*&#F$F'F$-F*6%7%F/F/F17$F.#\"\"(F/F2F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JdC)Q%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "arrange" }{TEXT -1 52 " will reconstruct the original series from t his sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sm3 := Sum(sqrt(2)*sin((2*n+1)*Pi/4)/n,n=1..infinity) ;\narrange(sm3,0.4388245731);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sm 3G-%$SumG6$*(\"\"##\"\"\"F)-%$sinG6#,$*(\"\"%!\"\",&*&F)F+%\"nGF+F+F+F +F+%#PiGF+F+F+F5F2/F5;F+%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #7K\"\"\"\"\"#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%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "arrange(sm3,0.4388245731,numterms=2 0,output=terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#77\"\"\"#!\"\"\"\" ##F&\"\"$#F$\"\"%#F$\"\"&#F&\"\"'#F&\"\"(#F$\"\")#F$\"\"*#F&\"#5#F&\"# 6#F$\"#7#F$\"#8#F&\"#9#F&\"#:#F$\"#;#F$\"#<#F&\"#=#F&\"#>#F$\"#?#F$\"# @" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We \+ can investigate rearrangements of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "arrange(sm3, -1/4,numterms=200);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^p\"\"!\"\"\"F %\"\"&F%\"\"$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(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(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "arrange(sm3,-1/4,numterms=20 ,output=terms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#77#!\"\"\"\"#\"\"\" #F%\"\"$#F%\"\"'#F%\"\"(#F%\"#5#F%\"#6#F'\"\"%#F%\"#9#F%\"#:#F%\"#=#F' \"\"&#F%\"#>#F%\"#A#F%\"#B#F%\"#E#F'\"\")#F%\"#F#F%\"#I#F%\"#J#F%\"#M " }}}{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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }