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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Slow growing divergent 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n=2..infinity)" "6#-%
$SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F);\"\"#%)infinityG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "We shall show that " }
{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n = 2 .. infinity)" "6#-%$SumG6$*&\"\"
\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F);\"\"#%)infinityG" }{TEXT -1 6 " is a
 " }{TEXT 259 34 "very slow growing divergent series" }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 11 "Note that  " }{XPPEDIT 18 0 "Limit(1/(n
*ln(n)),n=infinity)=0" "6#/-%&LimitG6$*&\"\"\"F(*&%\"nGF(-%#lnG6#F*F(!
\"\"/F*%)infinityG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Limit(1/(n*ln(n)),n=inf
inity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&\"
\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F)%)infinityG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We plot
 the graph of the function " }{XPPEDIT 18 0 "f(x)=1/(x*ln(x))" "6#/-%
\"fG6#%\"xG*&\"\"\"F)*&F'F)-%#lnG6#F'F)!\"\"" }{TEXT -1 87 " in order \+
to compare partial sums of the series with certain integrals of the fu
nction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
259 "f := x -> 1/(x*ln(x));\nrtbox := n -> [[n,0],[n+1,0],[n+1,f(n)],[
n,f(n)]]:\np1 := plot(f(x),x=1.5..8,thickness=2,color=red):\np2 := plo
ts[polygonplot]([seq(rtbox(n),n=2..8)],color=COLOR(RGB,.95,.6,.8)):\np
lots[display]([p1,p2],view=[1..8,0..1.7],tickmarks=[7,2]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&
\"\"\"F-*&9$F--%#lnG6#F/F-!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 
488 325 325 {PLOTDATA 2 "6'-%'CURVESG6%7Y7$$\"3++++++++:!#<$\"3Ea4D3B?
W;F*7$$\"3KL3_v.UN:F*$\"3Aa%*QNd%)=:F*7$$\"3km;/^2%3d\"F*$\"3RR=(\\=C'
49F*7$$\"3(**\\il7hig\"F*$\"3%o_-oNxOJ\"F*7$$\"3ILL3-:oT;F*$\"33l\\yO*
y(G7F*7$$\"3(*\\7yg'>Lq\"F*$\"3+-x&)o0N-6F*7$$\"3km\"z%>y&\\w\"F*$\"3'
GEUv`vG(**!#=7$$\"3))\\(oaPwU$=F*$\"37-OHTDj')*)FK7$$\"3OL$e9$\\f.>F*$
\"31\"GV.E8/;)FK7$$\"3gLL$e(HPt>F*$\"3'G1p4Qf\\X(FK7$$\"3hL$3--^J/#F*$
\"3w/-Zl4<]oFK7$$\"3ym\"HK#Q/#=#F*$\"3[o*H$p!)[teFK7$$\"3)H$e9J_\"3J#F
*$\"3A#f5+i<l;&FK7$$\"3(**\\7`c]TW#F*$\"3;I*zNZf!yXFK7$$\"3ULekeh/#e#F
*$\"3gO99%fFG3%FK7$$\"3)**\\i!>&*\\>FF*$\"3M%>IN1,bn$FK7$$\"3!om;a>()3
'GF*$\"3oQi9aiQDLFK7$$\"3_L$3x`@a)HF*$\"3)RQ(eel_iIFK7$$\"3K++D6xhDJF*
$\"3f<n'[?pt!GFK7$$\"3!****\\Pa*QmKF*$\"3'\\JR$\\0S'e#FK7$$\"3z***\\()
G\\?S$F*$\"3Jtt%*RJt+CFK7$$\"3.LekL8CDNF*$\"3uZS_\"zG9D#FK7$$\"3ammT!f
G<n$F*$\"3u)e\"H)oUR4#FK7$$\"3immm6D#ez$F*$\"3w;Vg,`,v>FK7$$\"3f*\\7GC
t,%RF*$\"39?JE,1(3&=FK7$$\"3Mmm;*4[z1%F*$\"3)4nTx>f>v\"FK7$$\"3O*\\7yO
P\"3UF*$\"3%HEo-*Qm`;FK7$$\"3d+vV.2jTVF*$\"3_/R@Tgso:FK7$$\"3]m;z%*p\"
4[%F*$\"3k$)488;'z[\"FK7$$\"3Zm\"H2HD)3YF*$\"3eTMCH(=+U\"FK7$$\"31L$3x
m\"zYZF*$\"3Qvu*)3Wj_8FK7$$\"37Leke**4!*[F*$\"3!*R(QaL*Q)G\"FK7$$\"3$)
*\\(=Z-&[,&F*$\"3787\")o6rO7FK7$$\"3mK$ek(Re\\^F*$\"3#>]&3'Hr[=\"FK7$$
\"3A****\\-rx)G&F*$\"3SOoc=J@N6FK7$$\"3m**\\i?/&\\U&F*$\"3Y30j$**z+4\"
FK7$$\"3Q*\\7y50nb&F*$\"3[MiU$HU$\\5FK7$$\"3?**\\PCi*Hq&F*$\"3O(Rl]0kr
+\"FK7$$\"3@mm;*HXW$eF*$\"3pyqR%>Pvr*!#>7$$\"3/++vV_zufF*$\"3q,6d!y\"3
j$*Ffw7$$\"3/Lek`J(>5'F*$\"3sS$\\=]p61*Ffw7$$\"3B++D,A,TiF*$\"3OV&\\4@
)H]()Ffw7$$\"3[l\"z%4r$=P'F*$\"3/^erx#GYZ)Ffw7$$\"3;+D1Moe3lF*$\"3&p7O
8><D?)Ffw7$$\"3slmT&o%GUmF*$\"3'*o()>k:5^zFfw7$$\"3E*\\7`%RD#y'F*$\"3m
f*z1W#=-xFfw7$$\"3mLLLy41<pF*$\"3kZG4)e>_Z(Ffw7$$\"3mK$3-k?\\0(F*$\"3A
eZ\"e'f6bsFfw7$$\"3Wm\"zWvQ;>(F*$\"3c8w=N3&z/(Ffw7$$\"3;++]-&osJ(F*$\"
3umCG<-nmoFfw7$$\"37m;/rVDhuF*$\"3q&\\O0dj)omFfw7$$\"3gKLL[q.!f(F*$\"3
M.>ek(f.]'Ffw7$$\"3r*\\7GCYts(F*$\"3r$Q/v=p)GjFfw7$$\"3#**\\(=i'o(eyF*
$\"3KSD![+E@<'Ffw7$$\"\")\"\"!$\"3\"zMPqLH7,'Ffw-%'COLOURG6&%$RGBG$\"*
++++\"!\")$Fe\\lFe\\lF_]l-%*THICKNESSG6#\"\"#-%)POLYGONSG6*7&7$$Fc]lFe
\\lF_]l7$$\"\"$Fe\\lF_]l7$F[^l$\"+0_Z8s!#57$Fi]lF^^l7&Fj]l7$$\"\"%Fe\\
lF_]l7$Fd^l$\"+a28MIF`^l7$F[^lFg^l7&Fc^l7$$\"\"&Fe\\lF_]l7$F\\_l$\"+,)
oL!=F`^l7$Fd^lF__l7&F[_l7$$\"\"'Fe\\lF_]l7$Fd_l$\"+p)pEC\"F`^l7$F\\_lF
g_l7&Fc_l7$$\"\"(Fe\\lF_]l7$F\\`l$\"+zP%=I*!#67$Fd_lF_`l7&F[`l7$Fc\\lF
_]l7$Fc\\l$\"+%*[STtFa`l7$F\\`lFf`l7&Fd`l7$$\"\"*Fe\\lF_]l7$F[al$\"+O$
H7,'Fa`l7$Fc\\lF^al-%&COLORG6&F[]l$\"#&*!\"#$Fe_l!\"\"$Fd\\lFhal-%*AXE
STICKSG6$F]`lFc]l-%+AXESLABELSG6$Q\"x6\"Q!Fabl-%%VIEWG6$;$\"\"\"Fe\\lF
c\\l;F_]l$\"#<Fhal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 38 "Partial sums satisfies the condition: " }}{PARA 256 "" 0 
"" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(1/(x*ln(x)),x = 2 .. n) < Sum(1
/(i*ln(i)),i = 2 .. n);" "6#2-%$IntG6$*&\"\"\"F(*&%\"xGF(-%#lnG6#F*F(!
\"\"/F*;\"\"#%\"nG-%$SumG6$*&F(F(*&%\"iGF(-F,6#F8F(F./F8;F1F2" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "d/dx" "6#
*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(ln(x))] =
 1/(x*ln(x));" "6#/7#-%#lnG6#-F&6#%\"xG*&\"\"\"F,*&F*F,-F&6#F*F,!\"\"
" }{TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "Int(1/(x*ln(x)),x=2..n)= ln(ln(x)" "6#/-%$IntG6$*&\"\"\"F(*&%\"xGF(
-%#lnG6#F*F(!\"\"/F*;\"\"#%\"nG-F,6#-F,6#F*" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([n,``],[2,``]) = ln(ln(n))-ln(ln(2))" "6#/-%*
PIECEWISEG6$7$%\"nG%!G7$\"\"#F),&-%#lnG6#-F.6#F(\"\"\"-F.6#-F.6#F+!\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 
18 0 "ln(ln(n))" "6#-%#lnG6#-F$6#%\"nG" }{TEXT -1 22 " tends to infini
ty as " }{TEXT 264 1 "n" }{TEXT -1 53 " tends to infinity, it follows \+
that the partial sums " }{XPPEDIT 18 0 "Sum(1/(i*ln(i)),i = 2 .. n);" 
"6#-%$SumG6$*&\"\"\"F'*&%\"iGF'-%#lnG6#F)F'!\"\"/F);\"\"#%\"nG" }
{TEXT -1 23 " also tend to infinity." }}{PARA 0 "" 0 "" {TEXT -1 16 "T
hus the series " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)),i = 2 .. infinity);" 
"6#-%$SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/%\"iG;\"\"#%)infinity
G" }{TEXT -1 22 " diverges to infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 64 "This analysis also provides a way of
 seeing how fast the series " }{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n=2..inf
inity)" "6#-%$SumG6$*&\"\"\"F'*&%\"nGF'-%#lnG6#F)F'!\"\"/F);\"\"#%)inf
inityG" }{TEXT -1 8 "  grows." }}{PARA 0 "" 0 "" {TEXT -1 35 "We can e
nsure that the partial sum " }{XPPEDIT 18 0 "Sum(1/(i*ln(i)),n=2..n)" 
"6#-%$SumG6$*&\"\"\"F'*&%\"iGF'-%#lnG6#F)F'!\"\"/%\"nG;\"\"#F/" }
{TEXT -1 31 " is greater than a given value " }{TEXT 265 1 "S" }{TEXT 
-1 27 " by ensuring that choosing " }{TEXT 266 1 "n" }{TEXT -1 9 " so \+
that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(exp(S+ln
(ln(2)))) < n" "6#2-%$expG6#-F%6#,&%\"SG\"\"\"-%#lnG6#-F-6#\"\"#F+%\"n
G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then  " }{XPPEDIT 
18 0 "ln(n) > exp(S+ln(ln(2))" "6#2-%$expG6#,&%\"SG\"\"\"-%#lnG6#-F+6#
\"\"#F)-F+6#%\"nG" }{TEXT -1 9 "   and   " }{XPPEDIT 18 0 "ln(ln(n)) >
 S+ln(ln(2))" "6#2,&%\"SG\"\"\"-%#lnG6#-F(6#\"\"#F&-F(6#-F(6#%\"nG" }
{TEXT -1 15 ", which gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "S < ln(ln(n))-ln(ln(2))" "6#2%\"SG,&-%#lnG6#-F'6#%\"nG
\"\"\"-F'6#-F'6#\"\"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(1/(x
*ln(x)),x = 2 .. n) < Sum(1/(i*ln(i)),i = 2 .. n)" "6#2-%$IntG6$*&\"\"
\"F(*&%\"xGF(-%#lnG6#F*F(!\"\"/F*;\"\"#%\"nG-%$SumG6$*&F(F(*&%\"iGF(-F
,6#F8F(F./F8;F1F2" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{TEXT 262 29 "_____________________________" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "For example, we
 can ensure that the sum is greater than 2.5 by choosing " }{TEXT 267 
1 "n" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "n >= 4649" "6#1\"%\\Y%\"
nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "evalf(exp(exp(2.5+ln(ln(2)))));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"+W4K[Y!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 35 "Actually, it is sufficient to take " 
}{XPPEDIT 18 0 "n = 245" "6#/%\"nG\"$X#" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum(1/(
i*ln(i)),i=2..245);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$S
umG6$*&\"\"\"F'*&%\"iGF'-%#lnG6#F)F'!\"\"/F);\"\"#\"$X#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+^h-+D!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 57 "We can ensure that the sum is greater t
han10 by choosing " }{TEXT 268 1 "n" }{TEXT -1 10 " so that  " }
{XPPEDIT 18 0 ".4235508745e6631 <= n;" "6#1-%&FloatG6$\"+X(3bB%\"%@m%
\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "evalf(exp(exp(10+ln(ln(2)))));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+X(3bB%\"%@m" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Estimatio
n of partial sums" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 153 "The following code sets up a function EM, using the Eule
r-Maclaurin summation formula, which can be used to calculate approxim
ations for the partial sums" }{XPPEDIT 18 0 " Sum(1/(i*ln(i)),i=2..n)
" "6#-%$SumG6$*&\"\"\"F'*&%\"iGF'-%#lnG6#F)F'!\"\"/F);\"\"#%\"nG" }
{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "k := 25:\ne
ulermac(1/((j+k)*ln(j+k)),j=2-k..n-k):\nEM := unapply(convert(%,polyno
m),n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#EMGf*6#%\"nG6\"6$%)operat
orG%&arrowGF(,fp*&#\"\"\"\"\"#F/*&F/F/*&9$F/-%#lnG6#F3F/!\"\"F/F/*&#F/
\"\")F/*&F/F/-F56#F:F7F/F/*&#F/\"\"(F/*&F/F/-F56#F@F7F/F/*&#F/\"\"&F/*
&F/F/-F56#FFF7F/F/*&#F/\"\"'F/*&F/F/-F56#FLF7F/F/*&F.F/*&F/F/-F56#F0F7
F/F/*&#F/\"\"$F/*&F/F/-F56#FVF7F/F/*&#F/\"\"%F/*&F/F/-F56#FfnF7F/F/*&#
F/\",+vVB:'F/*&F/F/*$)-F56#\"#DFLF/F7F/F/*&#\"%L_\"-++v$4Y#F/*&F/F/*$)
F`oFfnF/F7F/F7*&#F/\",+D\"y]?F/*&F/F/*$)F`oFFF/F7F/F/*&#\"*J;-Y#\".+]7
.d%=F/*&F/F/*$)F`oF0F/F7F/F/FRF7-F56#F4F/*&#F/\"#gF/*&F/F/*&)F3FfnF/)F
4FVF/F7F/F/*&#F/\"$_#F/*&F/F/*&)F3FLF/F4F/F7F/F7*&#F/\"$?\"F/*&F/F/*&F
^qF/F4F/F7F/F/*&#\"$L#\"+++voaF/*&F/F/*$)F`oFVF/F7F/F7*&#\"+F6MxC\"-+]
(o/B\"F/*&F/F/F`oF7F/F/-F56#FHF7*&#\"#6\"$?(F/*&F/F/*&F^qF/)F4F0F/F7F/
F/*&#F/\"#7F/*&F/F/*&)F3F0F/F_sF/F7F/F7*&#F/FbsF/*&F/F/*&FesF/F4F/F7F/
F7*&#\"$P\"\"&?^\"F/*&F/F/*&FeqF/F_sF/F7F/F7*&#FF\"$O$F/*&F/F/*&FeqF/F
_qF/F7F/F7*&#\"#<\"%35F/*&F/F/*&FeqF/)F4FfnF/F7F/F7*&#F/\"#%)F/*&F/F/*
&FeqF/)F4FFF/F7F/F7*&#F/FbqF/*&F/F/*&FeqF/)F4FLF/F7F/F7*&#F/FbsF/*&F/F
/-F56#FbsF7F/F/*&#F/\"#8F/*&F/F/-F56#F^vF7F/F/*&#F/\"#9F/*&F/F/-F56#Fd
vF7F/F/*&#F/\"#:F/*&F/F/-F56#FjvF7F/F/*&#F/\"#;F/*&F/F/-F56#F`wF7F/F/*
&#F/FgtF/*&F/F/-F56#FgtF7F/F/*&#F/\"#=F/*&F/F/-F56#F[xF7F/F/*&#F/\"#>F
/*&F/F/-F56#FaxF7F/F/*&#F/\"#?F/*&F/F/-F56#FgxF7F/F/*&#F/\"#@F/*&F/F/-
F56#F]yF7F/F/*&#F/\"#AF/*&F/F/-F56#FcyF7F/F/*&#F/\"#BF/*&F/F/-F56#FiyF
7F/F/*&#F/\"#CF/*&F/F/-F56#F_zF7F/F/*&FgqF/*&F/F/*&F^qF/F[uF/F7F/F/*&#
F/\"\"*F/*&F/F/-F56#FhzF7F/F/*&#F/\"#5F/*&F/F/-F56#F^[lF7F/F/*&#F/F[sF
/*&F/F/-F56#F[sF7F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 38 "Here are some examples for comparison." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "Sum(1/(i*ln(i)),i=2..1000)=evalf(sum(1/(i*ln(i)),i=2..1000));\neva
lf(EM(1000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*
&%\"iGF(-%#lnG6#F*F(!\"\"/F*;\"\"#\"%+5$\"+SdRFF!\"*" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"+[dRFF!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Sum(1/(i*ln(i)),i=2..10000)
=\n          evalf(evalhf(add(1/(i*ln(i)),i=2..10000)));\nevalf(EM(100
00));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*&%\"iGF(
-%#lnG6#F*F(!\"\"/F*;\"\"#\"&++\"$\"+!)3,:I!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+\")3,:I!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "Sum(1/(i*ln(i)),i=2..100000
)=\n          evalf(evalhf(sum(1/(i*ln(i)),i=2..100000)));\nevalf(EM(1
00000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*&%\"i
GF(-%#lnG6#F*F(!\"\"/F*;\"\"#\"'++5$\"+P%\\\"QK!\"*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+P%\\\"QK!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "Sum(1/(i*ln(i)),i=2..100000
0)=\n          evalf(evalhf(sum(1/(i*ln(i)),i=2..1000000)));\nevalf(EM
(1000000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*&%
\"iGF(-%#lnG6#F*F(!\"\"/F*;\"\"#\"(+++\"$\"+'fq/U$!\"*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+(fq/U$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Sum(1/(i*ln(i)),i=2..10000
000)=\n          evalf(evalhf(sum(1/(i*ln(i)),i=2..10000000)));\nevalf
(EM(10000000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F
(*&%\"iGF(-%#lnG6#F*F(!\"\"/F*;\"\"#\")+++5$\"+V7iuN!\"*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+V7iuN!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 37 "To obtain a sum near10 we need about " 
}{XPPEDIT 18 0 "10^4322;" "6#*$\"#5\"%AV" }{TEXT -1 7 " terms." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "evalf(EM(1e4322));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+j%=++\"!
\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "T
o obtain a sum near 20 we need about " }{XPPEDIT 18 0 "10^(10^8)" "6#)
\"#5*$F$\"\")" }{TEXT -1 7 " terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(EM(1e100000000));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&=R\\+#!\")" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 40 "Task: Clifford Pickover's infinity worms" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 83 "S
ee: Key's to Infinity, by Clifford A. Pickover, John Wiley & Sons, 199
5, page 153." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "On Callisto, a cold, icy moon of Jupiter, lives a race of intel
ligent worms called " }{TEXT 259 14 "infinity worms" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 185 "The number of years they are permitted \+
to live is based on mathematical formulas that they are free to select
 early in their lives. (The government uses this to prevent overpopula
tion.)" }}{PARA 0 "" 0 "" {TEXT -1 87 "The worms are particularly fasc
inated by divergent series, such as the harmonic series " }}{PARA 257 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/n,n = 1 .. infinity) = 1+
1/2+1/3+1/4+1/5+` . . . `;" "6#/-%$SumG6$*&\"\"\"F(%\"nG!\"\"/F);F(%)i
nfinityG,.F(F(*&F(F(\"\"#F*F(*&F(F(\"\"$F*F(*&F(F(\"\"%F*F(*&F(F(\"\"&
F*F(%(~.~.~.~GF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 126 "Thi
s series is rather slow growing in that it requires a large number of \+
terms even to produce a partial sum greater than 10. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 481 "An infinity worm make
s a selection of a divergent series, and is then allowed to live a num
ber of years equal to the minimum number of terms required to make the
 sum greater than 10. In other word, a very slow growing series is des
irable, because it allows a worm to live many years. ( In what follows
, it doesn't really matter whether you consider the years to be Earth \+
years or Jupiter years. It takes nearly 12 Earth years for Jupiter to \+
complete a single orbit around the sun.)" }}{PARA 0 "" 0 "" {TEXT -1 
194 "Naturally, the desire to discover slow growing divergent series g
ives the worm citizenry an incentive to become proficient in mathemati
cs, and therefore a highly intelligent society has evolved." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Six colorful wo
rms have recently selected their respective series as follows." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 10 "Green wo
rm" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT3D 
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{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/ln(ln(n)),n = 3
 .. infinity) = 1/ln(ln(3))+1/ln(ln(4))+1/ln(ln(5))+` . . . `;" "6#/-%
$SumG6$*&\"\"\"F(-%#lnG6#-F*6#%\"nG!\"\"/F.;\"\"$%)infinityG,**&F(F(-F
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0 "" {TEXT 256 9 "Blue worm" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/ln(n),n = 2 .. infinity) = 1/ln(2
)+1/ln(3)+1/ln(4)+` . . . `;" "6#/-%$SumG6$*&\"\"\"F(-%#lnG6#%\"nG!\"
\"/F,;\"\"#%)infinityG,**&F(F(-F*6#F0F-F(*&F(F(-F*6#\"\"$F-F(*&F(F(-F*
6#\"\"%F-F(%(~.~.~.~GF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 258 11 "Violet worm" }{TEXT -1 1 ":" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/sqrt(n),n = 1 .
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$*&\"\"\"F(-%%sqrtG6#%\"nG!\"\"/F,;F(%)infinityG,,F(F(*&F(F(-F*6#\"\"#
F-F(*&F(F(-F*6#\"\"$F-F(*&F(F(-F*6#\"\"%F-F(%(~.~.~.~GF(" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 12 "Cr
imson worm" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
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" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/n,n = 1 .. infinity) = 1+1/2+1/
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10 "Black worm" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(1/(n*ln(n)),n = 1 .. infinity) = 1/(2*ln(2))+1/(3*l
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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "(a) A
re all of these series divergent?" }}{PARA 0 "" 0 "" {TEXT -1 53 "(b) \+
How many years will each worm be allowed to live?" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "(c) Use the fact that " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG
\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(ln(ln(x)))] = 1/
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hich grows more slowly than any of these five series." }}{PARA 0 "" 0 
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