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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "The harmonic 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 20 "The harmonic series " }{XPPEDIT 18 0 "Sum(1/n,n = 1 .. infinity
)" "6#-%$SumG6$*&\"\"\"F'%\"nG!\"\"/F(;F'%)infinityG" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The ser
ies " }{XPPEDIT 18 0 "Sum(1/n,n = 1 .. infinity) = 1+1/2+1/3+1/4+1/5+1
/6+1/7+` . . . `;" "6#/-%$SumG6$*&\"\"\"F(%\"nG!\"\"/F);F(%)infinityG,
2F(F(*&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 15 " is called the " 
}{TEXT 261 15 "harmonic series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 
7 "leftbox" }{TEXT -1 8 " in the " }{MPLTEXT 1 0 7 "student" }{TEXT 
-1 49 " package is useful for investigating this series." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "p1 :
= student[leftbox](1/x,x=1..7,6,color=red,shading=COLOR(RGB,.95,.7,.9)
,\n     view=[0..7,0..2]):\np2 := plot(1/x,x=0.5..1,color=red,thicknes
s=2):\nplots[display]([p1,p2],view=[0..7,0..2]);\nSum(1/i,i=1..6);\nva
lue(%);\nevalf(%);" }}{PARA 13 "" 1 "" {GLPLOT2D 522 307 307 
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6F`p7$$\"3q**\\7.\"fF&))Fcp$\"3ror'zL\"fH6F`p7$$\"3Ymm;/Ogb*)Fcp$\"3b`
ZD>$>m6\"F`p7$$\"3y**\\ilAFj!*Fcp$\"3g-(eHDaL5\"F`p7$$\"3!HLLL)*pp;*Fc
p$\"3'>R8\\*H(34\"F`p7$$\"3kKL3xe,t#*Fcp$\"374l\\3yRy5F`p7$$\"3em;HdO=
y$*Fcp$\"3+'f#=gXIm5F`p7$$\"3))*****\\#>#[Z*Fcp$\"3eB8\"p!)Ga0\"F`p7$$
\"31mmT&G!e&e*Fcp$\"3QgJafOBV5F`p7$$\"3gKLL$)Qk%o*Fcp$\"3))Rmp*[iD.\"F
`p7$$\"37+]iSjE!z*Fcp$\"3\"[]l9nA9-\"F`p7$$\"35+]P40O\"*)*Fcp$\"3rQlvq
K)4,\"F`pF,Fg_lFa`l-%+AXESLABELSG6%Q\"x6\"Q!Fi_m-%%FONTG6#%(DEFAULTG-%
%VIEWG6$;F+Fgo;F+F." 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&
\"\"\"F'%\"iG!\"\"/F(;F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#
\\\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]C!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 46 "The picture illlustrates the 6 th parti
al sum " }{XPPEDIT 18 0 "S[6] = Sum(1/i,i = 1 .. 6);" "6#/&%\"SG6#\"\"
'-%$SumG6$*&\"\"\"F,%\"iG!\"\"/F-;F,F'" }{TEXT -1 53 "  as the sum of \+
the areas of the 6 shaded rectangles." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 46 "The six rectangles corresponding to t
he terms " }{XPPEDIT 18 0 "1+1/2+1/3+1/4+1/5+1/6;" "6#,.\"\"\"F$*&F$F$
\"\"#!\"\"F$*&F$F$\"\"$F'F$*&F$F$\"\"%F'F$*&F$F$\"\"&F'F$*&F$F$\"\"'F'
F$" }{TEXT -1 43 "  fit over the region bounded by the graph " }
{XPPEDIT 18 0 "y = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 6 ",
 the " }{TEXT 265 1 "x" }{TEXT -1 29 " axis and the vertical lines " }
{XPPEDIT 18 0 "x =1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "x = 7" "6#/%\"xG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 4 "Thus" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(1/
x,x = 1 .. 7) < Sum(1/i,i = 1 .. 6);" "6#2-%$IntG6$*&\"\"\"F(%\"xG!\"
\"/F);F(\"\"(-%$SumG6$*&F(F(%\"iGF*/F2;F(\"\"'" }{TEXT -1 2 "  " }}
{PARA 258 "" 0 "" {TEXT -1 6 "Since " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. 7) = ln(x);" "6#/-%$IntG6$*&\"\"\"
F(%\"xG!\"\"/F);F(\"\"(-%#lnG6#F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIE
CEWISE([7, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"(%!G7$\"\"\"F(" }
{XPPEDIT 18 0 "`` = ln(7)-ln(1);" "6#/%!G,&-%#lnG6#\"\"(\"\"\"-F'6#F*!
\"\"" }{XPPEDIT 18 0 "``= ln(7)" "6#/%!G-%#lnG6#\"\"(" }{TEXT -1 2 ", \+
" }}{PARA 258 "" 0 "" {TEXT -1 17 "it follows that  " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(7) < Sum(1/i,i = 1 .. 6);" "6#2
-%#lnG6#\"\"(-%$SumG6$*&\"\"\"F,%\"iG!\"\"/F-;F,\"\"'" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 
0 "Sum(1/i,i = 1 .. 6)=49/20" "6#/-%$SumG6$*&\"\"\"F(%\"iG!\"\"/F);F(
\"\"'*&\"#\\F(\"#?F*" }{XPPEDIT 18 0 "``=2.45" "6#/%!G-%&FloatG6$\"$X#
!\"#" }{TEXT -1 14 ", while ln(7) " }{TEXT 267 1 "~" }{TEXT -1 14 " 1.
945910149. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 83 "Sum(1/i,i=1..6)=sum(1/i,i=1..6);\n``=evalf(rhs(%));
\n'ln(7)'=evalf(evalf(ln(7),14));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%$SumG6$*&\"\"\"F(%\"iG!\"\"/F);F(\"\"'#\"#\\\"#?" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/%!G$\"++++]C!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%#lnG6#\"\"($\"+\\,\"f%>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
30 "If we take a partial sum with " }{TEXT 259 1 "n" }{TEXT -1 34 " te
rms, we obtain the inequality: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/x,x = 1 .. n+1) < Sum(1/i,i = 1 .. n);" "6#2-%$In
tG6$*&\"\"\"F(%\"xG!\"\"/F);F(,&%\"nGF(F(F(-%$SumG6$*&F(F(%\"iGF*/F3;F
(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 256 
"" 0 "" {XPPEDIT 18 0 "ln(n+1) < Sum(1/i,i = 1 .. n);" "6#2-%#lnG6#,&%
\"nG\"\"\"F)F)-%$SumG6$*&F)F)%\"iG!\"\"/F.;F)F(" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }
{XPPEDIT 18 0 "ln(N+1)" "6#-%#lnG6#,&%\"NG\"\"\"F(F(" }{TEXT -1 10 " t
ends to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 4 " as \+
" }{XPPEDIT 18 0 "N -> infinity" "6#f*6#%\"NG7\"6$%)operatorG%&arrowG6
\"%)infinityGF*F*F*" }{TEXT -1 25 ", we see that the series " }
{XPPEDIT 18 0 "Sum(1/i,i = 1 .. infinity);" "6#-%$SumG6$*&\"\"\"F'%\"i
G!\"\"/F(;F'%)infinityG" }{TEXT -1 23 "  diverges to infinity." }}
{PARA 0 "" 0 "" {TEXT -1 65 "Notice that the individual terms of the h
armonic series given by " }{XPPEDIT 18 0 "a[n]=1/n" "6#/&%\"aG6#%\"nG*
&\"\"\"F)F'!\"\"" }{TEXT -1 65 ", clearly tend to 0, so we have an exa
mple of a series for which " }{XPPEDIT 18 0 "Limit(a[n],n=infinity)=0
" "6#/-%&LimitG6$&%\"aG6#%\"nG/F*%)infinityG\"\"!" }{TEXT -1 6 ", but \+
" }{XPPEDIT 18 0 "Sum(a[n],n=1..infinity)=infinity" "6#/-%$SumG6$&%\"a
G6#%\"nG/F*;\"\"\"%)infinityGF." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 42 "Computations involving the harmonic ser
ies" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 14 "The procedure " }{MPLTEXT 1 0 3 "add" }{TEXT -1 49 " in t
he following code computes the partial sum  " }{XPPEDIT 18 0 "Sum(1/i,
i = 1 .. 1000000);" "6#-%$SumG6$*&\"\"\"F'%\"iG!\"\"/F(;F'\"(+++\"" }
{TEXT -1 146 "  of 1000000 terms of the harmonic series. Employing har
dware floating point arithmetic enables the calculation to be complete
d in a few seconds. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(evalhf(add(1/i,i=1..1000000)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sEFR9!\")" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }
{MPLTEXT 1 0 3 "sum" }{TEXT -1 36 " gives the result much more rapidly
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "i := 'i':\nevalf(evalf(sum(1/i,i=1..1000000),14));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sEFR9!\")" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }
{MPLTEXT 1 0 3 "sum" }{TEXT -1 75 " actually computes the finite sum b
y using a special mathematical function " }{XPPEDIT 18 0 "Psi" "6#%$Ps
iG" }{TEXT -1 31 " and the mathematical constant " }{XPPEDIT 18 0 "gam
ma" "6#%&gammaG" }{TEXT -1 32 ", which we shall discuss later. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "Sum(1/i,i=1..n)=sum(1/i,i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%$SumG6$*&\"\"\"F(%\"iG!\"\"/F);F(%\"nG,&-%$PsiG6#,&F-F(F(F(F(%&ga
mmaGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The following comput
ations show that the partial sums of the harmonic series " }{XPPEDIT 
18 0 "Sum(1/n,n=1..infinity)" "6#-%$SumG6$*&\"\"\"F'%\"nG!\"\"/F(;F'%)
infinityG" }{TEXT -1 19 "  grow very slowly." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "H := n -> [Sum(1/
i,i=1..n),`=`,evalf(sum(1./i,i=1..n))]:\nmatrix([H(1),H(2),H(3),[`:`,`
:`,`:`],H(100),[`:`,`:`,`:`],\nH(10^4),[`:`,`:`,`:`],H(10^6),[`:`,`:`,
`:`],H(10^8),[`:`,`:`,`:`],\nH(10^16),[`:`,`:`,`:`],H(10^32)]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#717%-%$SumG6$*&\"\"\"F,%\"
iG!\"\"/F-;F,F,%\"=G$F,\"\"!7%-F)6$F+/F-;F,\"\"#F1$\"+++++:!\"*7%-F)6$
F+/F-;F,\"\"$F1$\"+LLLL=F<7%%\":GFFFF7%-F)6$F+/F-;F,\"$+\"F1$\"+=vP(=&
F<FE7%-F)6$F+/F-;F,\"&++\"F1$\"+Ogg(y*F<FE7%-F)6$F+/F-;F,\"(+++\"F1$\"
+sEFR9!\")FE7%-F)6$F+/F-;F,\"*++++\"F1$\"+T'*y**=FinFE7%-F)6$F+/F-;F,
\"2++++++++\"F1$\"+:x&=u$FinFE7%-F)6$F+/F-;F,\"B++++++++++++++++\"F1$
\"+kQ*fU(Fin" }}}{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 41 "In the previous section we showed that   " }
{XPPEDIT 18 0 "ln(N+1)<Sum(1/i,i=1..N)" "6#2-%#lnG6#,&%\"NG\"\"\"F)F)-
%$SumG6$*&F)F)%\"iG!\"\"/F.;F)F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "It is straightforward to
 make an upper estimate for the number terms of the harmonic series re
quired to make the partial sum larger than any specified positive numb
er." }}{PARA 0 "" 0 "" {TEXT -1 73 "For example, if we want to obtain \+
a sum larger than 100, we could choose " }{TEXT 266 1 "N" }{TEXT -1 9 
" so that " }{XPPEDIT 18 0 "100 <= ln(N+1);" "6#1\"$+\"-%#lnG6#,&%\"NG
\"\"\"F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 " This gives
 " }{XPPEDIT 18 0 "exp(100)-1 <= N;" "6#1,&-%$expG6#\"$+\"\"\"\"F)!\"
\"%\"NG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(exp(100)-1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+Ur6)o#\"#M" }}}{PARA 0 "" 0 "" {TEXT -1 30 "Le
t's take the sum of around  " }{XPPEDIT 18 0 "3*`.`*10^43;" "6#*(\"\"$
\"\"\"%\".GF%\"#5\"#V" }{TEXT -1 7 " terms." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Sum(1/i,i=1..3*10^
43);\nevalf(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&
\"\"\"F'%\"iG!\"\"/F(;F'\"M+++++++++++++++++++++I" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+q)po+\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 58 "In order to obtain a sum larger than 200,
 we could choose " }{TEXT 268 1 "N" }{TEXT -1 9 " so that " }{XPPEDIT 
18 0 "200 <= ln(N+1);" "6#1\"$+#-%#lnG6#,&%\"NG\"\"\"F*F*" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 " This gives " }{XPPEDIT 18 0 "exp
(200)-1 <= N;" "6#1,&-%$expG6#\"$+#\"\"\"F)!\"\"%\"NG" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "evalf(exp(200)-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+oP(f
A(\"#x" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 
4 "Note" }{TEXT -1 89 ": The number of electrons, protons and neutrons
 in the universe is estimated to be about " }{XPPEDIT 18 0 "10^79" "6#
*$\"#5\"#z" }{TEXT -1 47 ", so we certainly need a large number of ter
ms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "Sum(1/i,i=1..0.7226e87);\nevalf(value(%));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'%\"iG!\"\"/F(;F'$\"%Es\"#
$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$>sd+#!\"(" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "In order to obtain a s
um larger than " }{XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 18 ",
 we could choose " }{TEXT 269 1 "N" }{TEXT -1 9 " so that " }{XPPEDIT 
18 0 "10^6 <= ln(N+1);" "6#1*$\"#5\"\"'-%#lnG6#,&%\"NG\"\"\"F,F," }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 " This gives " }{XPPEDIT 
18 0 "exp(10^6)-1 <= N;" "6#1,&-%$expG6#*$\"#5\"\"'\"\"\"F+!\"\"%\"NG
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "evalf(exp(10^6)-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+(R:K.$\"'&GM%" }}}{PARA 0 "" 0 "" {TEXT -1 11 "  Th
is is  " }{XPPEDIT 18 0 ".3033215397*10^434295" "6#*&-%&FloatG6$\"+(R:
K.$!#5\"\"\"*$\"#5\"'&HM%F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum(1/i,i=1..0.304
e434295);\nevalf(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG
6$*&\"\"\"F'%\"iG!\"\"/F(;F'$\"$/$\"'#HM%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+z0++5!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 17 "Euler's constant " }{XPPEDIT 18 0 "gamma" "6#%&gammaG" 
}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "The curve in the follow
ing picture is " }{XPPEDIT 18 0 "y = 1/x" "6#/%\"yG*&\"\"\"F&%\"xG!\"
\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 380 "rtbox := n -> [[n,0],[n+1,0],[n+1,1/(n+1)],
[n,1/(n+1)]]:\nltbox := n -> [[n,1/(n+1)],[n+1,1/(n+1)],[n+1,1/n],[n,1
/n]]:\np1 := plot(1/x,x=0.5..7,thickness=2,color=red):\np2 := plots[po
lygonplot]([seq(ltbox(n),n=1..6)],color=COLOR(RGB,.95,.6,.8)):\np3 := \+
plots[polygonplot]([seq(rtbox(n),n=1..6)],color=COLOR(RGB,.7,.95,.7)):
\nplots[display]([p1,p2,p3],view=[0..7,0..2],tickmarks=[7,3]);" }}
{PARA 13 "" 1 "" {GLPLOT2D 518 303 303 {PLOTDATA 2 "6(-%'CURVESG6%7Y7$
$\"3++++++++]!#=$\"\"#\"\"!7$$\"3EL$3_v.UN&F*$\"3#>xxDM\"pn=!#<7$$\"3]
mmT5vS3dF*$\"3g2<o`?!=v\"F37$$\"3u**\\il7higF*$\"3?M`\\jUX\\;F37$$\"3*
HLL3-:oT'F*$\"31jL!z_0%e:F37$$\"3n*\\7yg'>LqF*$\"36mQr(eG=U\"F37$$\"3M
m;z%>y&\\wF*$\"3vm4>A;E28F37$$\"3)**\\(oaPwU$)F*$\"3e&[ZVbV')>\"F37$$
\"3hLLe9$\\f.*F*$\"3M1%\\8e!p16F37$$\"3wLLLe(HPt*F*$\"35O.`<aNF5F37$$
\"3RL$3--^J/\"F3$\"3yo\"eMmRje*F*7$$\"3cm\"HK#Q/#=\"F3$\"3%oxL3BB*f%)F
*7$$\"3?Le9J_\"3J\"F3$\"3>!*oH`-%)GwF*7$$\"3(**\\7`c]TW\"F3$\"3WVc#Gu&
[CpF*7$$\"3?Lekeh/#e\"F3$\"3+i%fv3G4K'F*7$$\"3)**\\i!>&*\\><F3$\"3=fd>
4dk:eF*7$$\"3emmT&>()3'=F3$\"3u9Sdj3yt`F*7$$\"3IL$3x`@a)>F3$\"3-I9*3x8
n.&F*7$$\"3K++D6xhD@F3$\"3J54g3[^/ZF*7$$\"3!****\\Pa*QmAF3$\"3ua\\naTI
7WF*7$$\"3z***\\()G\\?S#F3$\"3'o&****3>6jTF*7$$\"3.LekL8CDDF3$\"3;JxI]
v,gRF*7$$\"3ammT!fG<n#F3$\"3?vHHc^*Gu$F*7$$\"3immm6D#ez#F3$\"3EUs5i\\w
wNF*7$$\"3f*\\7GCt,%HF3$\"3%H6Dq7g6S$F*7$$\"3Mmm;*4[z1$F3$\"3Cf(GDb2&f
KF*7$$\"3\")*\\7yOP\"3KF3$\"3&)=!R#3N2<JF*7$$\"37+vV.2jTLF3$\"3#)z]E44
b#*HF*7$$\"3]m;z%*p\"4[$F3$\"3mt0u@i!G(GF*7$$\"3Zm\"H2HD)3OF3$\"3Q$GmE
z%)4x#F*7$$\"31L$3xm\"zYPF3$\"3#Gr655]*oEF*7$$\"37Leke**4!*QF3$\"3YW)[
<.G1d#F*7$$\"3$)*\\(=Z-&[,%F3$\"33k?%e#Hv!\\#F*7$$\"3mK$ek(Re\\TF3$\"3
\"4/&fO,))4CF*7$$\"3A****\\-rx)G%F3$\"3#f!Rh!*pmJBF*7$$\"3m**\\i?/&\\U
%F3$\"3]rGcEB\"*fAF*7$$\"3Q*\\7y50nb%F3$\"3w9WEs\"oX>#F*7$$\"3?**\\PCi
*Hq%F3$\"3@:\"=s1/j7#F*7$$\"3@mm;*HXW$[F3$\"3c(e#fO'*[o?F*7$$\"3/++vV_
zu\\F3$\"3)\\9qB)H85?F*7$$\"3/Lek`J(>5&F3$\"3_T,k$*f-g>F*7$$\"3B++D,A,
T_F3$\"3!H(3?w$G!3>F*7$$\"3[l\"z%4r$=P&F3$\"3m$4.=ag:'=F*7$$\"3;+D1Moe
3bF3$\"3[4<A?wM:=F*7$$\"3slmT&o%GUcF3$\"3E#G8&)pJBx\"F*7$$\"3E*\\7`%RD
#y&F3$\"3d!Hl\\SH%H<F*7$$\"3mLLLy41<fF3$\"3MZ#\\W@G+p\"F*7$$\"3mK$3-k?
\\0'F3$\"3A?954$\\:l\"F*7$$\"3Wm\"zWvQ;>'F3$\"3eZG0083:;F*7$$\"3;++]-&
osJ'F3$\"3a>e$ejiHe\"F*7$$\"37m;/rVDhkF3$\"3uay#Q4(oZ:F*7$$\"3gKLL[q.!
f'F3$\"3WL>%>:Uu^\"F*7$$\"3r*\\7GCYts'F3$\"3+@5OU-Z'[\"F*7$$\"3#**\\(=
i'o(eoF3$\"3'f)RY\\w)zX\"F*7$$\"\"(F-$\"3\\G9dG9dG9F*-%'COLOURG6&%$RGB
G$\"*++++\"!\")$F-F-F^]l-%*THICKNESSG6#F,-%)POLYGONSG6)7&7$$\"\"\"F-$
\"+++++]!#57$F+Fi]l7$F+Fg]l7$Fg]lFg]l7&7$F+$\"+LLLLLF[^l7$$\"\"$F-Fa^l
7$Fd^lFi]lF\\^l7&7$Fd^l$\"+++++DF[^l7$$\"\"%F-Fi^l7$F\\_lFa^lFc^l7&7$F
\\_l$\"+++++?F[^l7$$\"\"&F-Fa_l7$Fd_lFi^lF[_l7&7$Fd_l$\"+nmmm;F[^l7$$
\"\"'F-Fi_l7$F\\`lFa_lFc_l7&7$F\\`l$\"+H9dG9F[^l7$Fc\\lFa`l7$Fc\\lFi_l
F[`l-%&COLORG6&Fj\\l$\"#&*!\"#$F]`l!\"\"$\"\")F\\al-Fc]l6)7&7$Fg]lF^]l
7$F+F^]lF\\^lFf]l7&Fcal7$Fd^lF^]lFc^lF`^l7&Feal7$F\\_lF^]lF[_lFh^l7&Fg
al7$Fd_lF^]lFc_lF`_l7&Fial7$F\\`lF^]lF[`lFh_l7&F[bl7$Fc\\lF^]lFc`lF``l
-Ff`l6&Fj\\l$Fd\\lF\\alFh`lF`bl-%*AXESTICKSG6$Fd\\lFe^l-%+AXESLABELSG6
$Q\"x6\"Q!Fhbl-%%VIEWG6$;F^]lFc\\l;F^]lF+" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "In the first section a similar \+
picture was used to suggest that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "Int(1/x,x = 1 .. n+1) = ``;" "6#/-%$IntG6$*&\"\"\"F(%
\"xG!\"\"/F);F(,&%\"nGF(F(F(%!G" }{XPPEDIT 18 0 "ln(n+1) < Sum(1/i,i =
 1 .. n);" "6#2-%#lnG6#,&%\"nG\"\"\"F)F)-%$SumG6$*&F)F)%\"iG!\"\"/F.;F
)F(" }{TEXT -1 6 ", for " }{XPPEDIT 18 0 "n >=1" "6#1\"\"\"%\"nG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "The infinite sum of the
 areas cut off from the rectangles " }{TEXT 261 15 "above the curve" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "y=1/x" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }
{TEXT -1 18 "  turns out to be " }{TEXT 261 6 "finite" }{TEXT -1 1 ".
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/x,x=n..n+1)
=ln(x)" "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"/F);%\"nG,&F-F(F(F(-%#lnG6#F)
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([n+1,``],[n,``]) = ln(n+1)
-ln(n)" "6#/-%*PIECEWISEG6$7$,&%\"nG\"\"\"F*F*%!G7$F)F+,&-%#lnG6#,&F)F
*F*F*F*-F/6#F)!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "ln((n+1)/n)" "6
#-%#lnG6#*&,&%\"nG\"\"\"F)F)F)F(!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 45 "so this total area is the sum of the series  " }
{XPPEDIT 18 0 "Sum(1/n-ln(n+1)+ln(n),n = 1 .. infinity) = Sum(1/n-ln((
n+1)/n),n = 1 .. infinity);" "6#/-%$SumG6$,(*&\"\"\"F)%\"nG!\"\"F)-%#l
nG6#,&F*F)F)F)F+-F-6#F*F)/F*;F)%)infinityG-F%6$,&*&F)F)F*F+F)-F-6#*&,&
F*F)F)F)F)F*F+F+/F*;F)F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "We can establish the convergence \+
of this series by comparison with the series whose terms are the areas
 of the pink rectangles. " }}{PARA 0 "" 0 "" {TEXT -1 16 "This series \+
is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/n-1/(n+
1),n = 1 .. infinity) = Sum(1/(n*(n+1)),n = 1 .. infinity);" "6#/-%$Su
mG6$,&*&\"\"\"F)%\"nG!\"\"F)*&F)F),&F*F)F)F)F+F+/F*;F)%)infinityG-F%6$
*&F)F)*&F*F),&F*F)F)F)F)F+/F*;F)F0" }{TEXT -1 6 " = 1. " }}{PARA 0 "" 
0 "" {TEXT -1 31 "The partial sums of the series " }{XPPEDIT 18 0 "Sum
(1/n-ln((n+1)/n),n = 1 .. infinity)" "6#-%$SumG6$,&*&\"\"\"F(%\"nG!\"
\"F(-%#lnG6#*&,&F)F(F(F(F(F)F*F*/F);F(%)infinityG" }{TEXT -1 16 " have
 the form  " }{XPPEDIT 18 0 "Sum(1/i,i = 1 .. n)-ln(n+1);" "6#,&-%$Sum
G6$*&\"\"\"F(%\"iG!\"\"/F);F(%\"nGF(-%#lnG6#,&F-F(F(F(F*" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 46 "We tabulate some values of these pa
rtial sums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 236 "H := n->[`10`^n,evalf(sum(1/i,i=1..10^n)),evalf(ln
(10^n+1)),\nevalf(evalf(sum(1/i,i=1..10^n)-ln(10^n+1),15))]:\nmatrix([
[n,Sum(1/i,i=1..n),ln(n+1),Sum(1/i,i=1..n)-ln(n+1)],\nH(0),H(1),H(2),H
(3),H(4),H(5),H(6),H(7),H(8),H(9),H(10),H(100)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7/7&%\"nG-%$SumG6$*&\"\"\"F-%\"iG!\"\"/F.;F
-F(-%#lnG6#,&F(F-F-F-,&F)F-F2F/7&F-$F-\"\"!$\"+1=ZJp!#5$\"+%>G&oIF<7&%
#10G$\"+a#o*GH!\"*$\"+t_*yR#FC$\"+7)H2J&F<7&*$)F@\"\"#F-$\"+=vP(=&FC$
\"+<07:YFC$\"+3+dAdF<7&*$)F@\"\"$F-$\"+h3Z&[(FC$\"+zZv3pFC$\"+73;ndF<7
&*$)F@\"\"%F-$\"+Ogg(y*FC$\"+n.W5#*FC$\"+\"pc;x&F<7&*$)F@\"\"&F-$\"+7Y
,47!\")$\"+YNH^6Ffo$\"+\\m5sdF<7&*$)F@\"\"'F-$\"+sEFR9Ffo$\"+c6b\"Q\"F
fo$\"+\\;:sdF<7&*$)F@\"\"(F-$\"+O6`p;Ffo$\"+v&4=h\"Ffo$\"+\\h:sdF<7&*$
)F@\"\")F-$\"+T'*y**=Ffo$\"+v!o?%=Ffo$\"+*fc@x&F<7&*$)F@\"\"*F-$\"+]\"
[+8#Ffo$\"+%eEB2#Ffo$\"+Wm:sdF<7&*$)F@\"#5F-$\"+fmIgBFfo$\"+$4&e-BFfo$
\"+\\m:sdF<7&*$)F@\"$+\"F-$\"+]sN3B!\"($FjrFcsF[s" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The limit of this sequenc
e is " }{TEXT 261 16 "Euler's constant" }{TEXT -1 2 "  " }{XPPEDIT 18 
0 "gamma" "6#%&gammaG" }{TEXT -1 1 " " }{TEXT 260 1 "~" }{TEXT -1 14 "
 0.5772156649." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 95 "Limit(Sum(1/i,i=1..n)-ln(n+1),n=infinity);\nLimi
t(sum(1/i,i=1..n)-ln(n+1),n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%&LimitG6$,&-%$SumG6$*&\"\"\"F+%\"iG!\"\"/F,;F+%\"nGF+
-%#lnG6#,&F0F+F+F+F-/F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%&LimitG6$,(-%$PsiG6#,&%\"nG\"\"\"F,F,F,%&gammaGF,-%#lnGF)!\"\"/F+%)i
nfinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&gammaG" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum(1/n-l
n(n+1)+ln(n),n=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$SumG6$,(*&\"\"\"F(%\"nG!\"\"F(-%#lnG6#,&F)F(F(F(F*-F,6#F)F(/F
);F(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&gammaG" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ev
alf(gamma,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5hgG`,\\m:sd!#?" 
}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "In fa
ct the psi function " }{XPPEDIT 18 0 "Psi" "6#%$PsiG" }{TEXT -1 57 ", \+
which appears above, is defined for a positive integer " }{TEXT 270 1 
"n" }{TEXT -1 3 " by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Psi(n) = Sum(1/i,i = 1 .. n-1)-gamma;" "6#/-%$PsiG6#%\"nG,&-%$SumG6
$*&\"\"\"F-%\"iG!\"\"/F.;F-,&F'F-F-F/F-%&gammaGF/" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 17 "so, for example, " }{XPPEDIT 18 0 "Psi(1)
=-gamma" "6#/-%$PsiG6#\"\"\",$%&gammaG!\"\"" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ev
alf(Psi(1),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!5hgG`,\\m:sd!#?" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }
{TEXT 264 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the series \+
" }{XPPEDIT 18 0 "Sum(3/(5*n+7),n=1..infinity)" "6#-%$SumG6$*&\"\"$\"
\"\",&*&\"\"&F(%\"nGF(F(\"\"(F(!\"\"/F,;F(%)infinityG" }{TEXT -1 23 " \+
 diverges to infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "
" 0 "" {TEXT 263 8 "Solution" }{TEXT 271 4 ":   " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "When " }{XPPEDIT 18 0 "7 <= n" "6#1\"\"(%\"nG" }{TEXT -1 
4 ",   " }{XPPEDIT 18 0 "1/(2*n)" "6#*&\"\"\"F$*&\"\"#F$%\"nGF$!\"\"" 
}{TEXT -1 4 " =  " }{XPPEDIT 18 0 "3/(5*n+7) >= 3/(6*n)" "6#1*&\"\"$\"
\"\"*&\"\"'F&%\"nGF&!\"\"*&F%F&,&*&\"\"&F&F)F&F&\"\"(F&F*" }{TEXT -1 
2 " ." }}{PARA 0 "" 0 "" {TEXT -1 7 "Since  " }{XPPEDIT 18 0 "Sum(1/(2
*n),n = 7 .. infinity);" "6#-%$SumG6$*&\"\"\"F'*&\"\"#F'%\"nGF'!\"\"/F
*;\"\"(%)infinityG" }{TEXT -1 31 " diverges to infinity, so does " }
{XPPEDIT 18 0 "Sum(3/(5*n+7),n = 7 .. infinity)" "6#-%$SumG6$*&\"\"$\"
\"\",&*&\"\"&F(%\"nGF(F(\"\"(F(!\"\"/F,;F-%)infinityG" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "Sum(3/(5*n+7),n=1..infinity);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$SumG6$,$*&\"\"\"F(,&%\"nG\"\"&\"\"(F(!\"\"\"\"$/F*;F
(%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question
" }{TEXT 272 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 21 "Show that the ser
ies " }{XPPEDIT 18 0 "Sum(2/((n+1)*(n+3)),n=1..infinity)" "6#-%$SumG6$
*&\"\"#\"\"\"*&,&%\"nGF(F(F(F(,&F+F(\"\"$F(F(!\"\"/F+;F(%)infinityG" }
{TEXT -1 28 " converges and find its sum." }}{PARA 257 "" 0 "" {TEXT 
263 8 "Solution" }{TEXT 273 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "2/((n+1)*(n+3))<=2/n^2" "6#1*&\"\"#\"\"\"*&,&%\"nGF&F&
F&F&,&F)F&\"\"$F&F&!\"\"*&F%F&*$F)F%F," }{TEXT -1 7 "  for  " }
{XPPEDIT 18 0 "n>=1" "6#1\"\"\"%\"nG" }{TEXT -1 4 ", so" }}{PARA 0 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Sum(2/((n+1)*(n+3)),n = 1 .. infi
nity)" "6#-%$SumG6$*&\"\"#\"\"\"*&,&%\"nGF(F(F(F(,&F+F(\"\"$F(F(!\"\"/
F+;F(%)infinityG" }{TEXT -1 30 " converges by comparison with " }
{XPPEDIT 18 0 "Sum(2/n^2,n = 1 .. infinity)=2*Sum(1/n^2,n = 1 .. infin
ity)" "6#/-%$SumG6$*&\"\"#\"\"\"*$%\"nGF(!\"\"/F+;F)%)infinityG*&F(F)-
F%6$*&F)F)*$F+F(F,/F+;F)F/F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Pi^2/3
" "6#*&%#PiG\"\"#\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "To find the sum of the series n
ote that the general term has the partial fraction expansion " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2/((n+1)*(n+3))=1/(n+
1)-1/(n+3)" "6#/*&\"\"#\"\"\"*&,&%\"nGF&F&F&F&,&F)F&\"\"$F&F&!\"\",&*&
F&F&,&F)F&F&F&F,F&*&F&F&,&F)F&F+F&F,F," }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "convert
(2/((n+1)*(n+3)),parfrac,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"
\"\"F%,&%\"nGF%F%F%!\"\"F%*&F%F%,&F'F%\"\"$F%F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 9 "We have: " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Sum(1/(i+1)-1/(i+3),i = 1 .. n) = ``(1/2-1/4)+``(1/3-1/
5)+``(1/4-1/6)+` . . . `+``(1/n-1/(n+2))+``(1/(n+1)-1/(n+3))" "6#/-%$S
umG6$,&*&\"\"\"F),&%\"iGF)F)F)!\"\"F)*&F)F),&F+F)\"\"$F)F,F,/F+;F)%\"n
G,.-%!G6#,&*&F)F)\"\"#F,F)*&F)F)\"\"%F,F,F)-F56#,&*&F)F)F/F,F)*&F)F)\"
\"&F,F,F)-F56#,&*&F)F)F;F,F)*&F)F)\"\"'F,F,F)%(~.~.~.~GF)-F56#,&*&F)F)
F2F,F)*&F)F),&F2F)F9F)F,F,F)-F56#,&*&F)F),&F2F)F)F)F,F)*&F)F),&F2F)F/F
)F,F,F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "This partial \+
sum \"telescopes\" to " }{XPPEDIT 18 0 "1/2+1/3-1/(n+2)-1/(n+3) = 5/6-
(2*n+5)/((n+2)*(n+3)" "6#/,**&\"\"\"F&\"\"#!\"\"F&*&F&F&\"\"$F(F&*&F&F
&,&%\"nGF&F'F&F(F(*&F&F&,&F-F&F*F&F(F(,&*&\"\"&F&\"\"'F(F&*&,&*&F'F&F-
F&F&F2F&F&*&,&F-F&F'F&F&,&F-F&F*F&F&F(F(" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sum(1
/(i+1)-1/(i+3),i=1..n);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,&*&,&\"\"&\"\"\"%\"nG\"\"#F'*&,&F(F'\"\"$F'F',&F(F'F)F'F'!\"\"F.#F&\"
\"'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"&\"\"\"%\"nG\"\"#F'
*&,&F(F'\"\"$F'F',&F(F'F)F'F'!\"\"F.#F&\"\"'F'" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The limit of the sequence
 of partial sums as n tends to infinity is " }{XPPEDIT 18 0 "5/6" "6#*
&\"\"&\"\"\"\"\"'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Sum(2/((n+1)*(n+3)),n=1.
.infinity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*
&\"\"\"F(*&,&%\"nGF(F(F(F(,&F+F(\"\"$F(F(!\"\"\"\"#/F+;F(%)infinityG" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"'" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Note that the argument u
sed to find the sum of the series also establishes the convergence ind
ependently of the first method." }}{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 "" }}{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 }