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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 257 
"" 0 "" {TEXT -1 37 "Lesson 14:  Introduction to Sequences" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 9 "Sequences" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 2 "A " }{TEXT 256 8 "sequence" }{TEXT -1 184 " is simply an i
nfinite list; that is an infinite collection of objects arranged in so
me order.  In this course, we will deal almost exclusively with sequen
ces of numbers.  For example," }}{PARA 256 "" 0 "" {TEXT -1 24 "1, 2, \+
3, 4, 5, 6, 7, ..." }}{PARA 256 "" 0 "" {TEXT -1 38 "1, -1/3, 1/9, -1/
27, 1/81, -1/243, ..." }}{PARA 256 "" 0 "" {TEXT -1 33 "1/2, 2/3, 3/4,
 4/5, 5/6, 6/7, ..." }}{PARA 0 "" 0 "" {TEXT -1 109 "are three sequenc
es of numbers.  (Note the ... symbols, which indicate that the list co
ntinues indefinitely.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 238 "We typically denote a sequence by a single letter; \+
the individual terms in the sequence are denoted by the same letter, w
ith a subscript showing the position of the term in the sequence.  For
 example, if the last sequence above is called " }{XPPEDIT 18 0 "a;" "
6#%\"aG" }{TEXT -1 6 ", then" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "a_1 = 1
/2;" "6#/%$a_1G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 4 " ,  " }{XPPEDIT 18 
0 "a_2 = 2/3;" "6#/%$a_2G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 4 " ,  " }
{XPPEDIT 18 0 "a_3 = 3/4;" "6#/%$a_3G*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 
-1 18 "  and so on.  The " }{TEXT 257 12 "general term" }{TEXT -1 5 ",
 or " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 1 "-" }{TEXT 258 7 "th t
erm" }{TEXT -1 26 ", is given by the formula " }{XPPEDIT 18 0 "a_n = n
/(n+1);" "6#/%$a_nG*&%\"nG\"\"\",&F&F'F'F'!\"\"" }{TEXT -1 2 " ." }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 1" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 54 "What is the general term in the second sequence abov
e?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 273 9 "Solution." }{TEXT -1 2 "  " }{XPPEDIT 18 0 "(-1/3)^(n+1);
" "6#),$*&\"\"\"F&\"\"$!\"\"F(,&%\"nGF&F&F&" }{TEXT -1 25 " .  (Why is
 the exponent " }{XPPEDIT 18 0 "n+1;" "6#,&%\"nG\"\"\"F%F%" }{TEXT -1 
14 " and not just " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 2 "?)" }}}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "One way to produce a list in " }
{TEXT 261 5 "Maple" }{TEXT -1 28 " is to use the dollar sign, " }
{TEXT 262 1 "$" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "n^2 $n=1..10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"%\"
\"*\"#;\"#D\"#O\"#\\\"#k\"#\")\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "n/(n+1) $n=3..7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'#
\"\"$\"\"%#F%\"\"&#F'\"\"'#F)\"\"(#F+\"\")" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 61 "Note the use of the word 'list' rather than 'sequence' he
re; " }{TEXT 263 5 "Maple" }{TEXT -1 164 " is only producing a finite \+
number of terms, not the whole sequence.  This is enough, though,  to \+
let us get a picture of the sequence by plotting the general term " }
{XPPEDIT 18 0 "a_n;" "6#%$a_nG" }{TEXT -1 18 " as a function of " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 28 ".  With the correct syntax,
 " }{TEXT 264 6 " Maple" }{TEXT -1 3 "'s " }{TEXT 260 4 "plot" }{TEXT 
-1 114 " command will do this for us.  In the examples below, we first
 define a list of points to be plotted (the points (" }{XPPEDIT 18 0 "
n;" "6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a_n;" "6#%$a_nG" }
{TEXT -1 27 ")), then pass this list to " }{TEXT 259 4 "plot" }{TEXT 
-1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a := [[n, n^2] \+
$n=1..10];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7,7$\"\"\"F'7$\"\"
#\"\"%7$\"\"$\"\"*7$F*\"#;7$\"\"&\"#D7$\"\"'\"#O7$\"\"(\"#\\7$\"\")\"#
k7$F-\"#\")7$\"#5\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
plot(a, x=0..12, style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6&-%'CURVESG6$7,7$$\"\"\"\"\"!F(7$$\"\"#F*$\"\"%F*7$$
\"\"$F*$\"\"*F*7$F.$\"#;F*7$$\"\"&F*$\"#DF*7$$\"\"'F*$\"#OF*7$$\"\"(F*
$\"#\\F*7$$\"\")F*$\"#kF*7$F3$\"#\")F*7$$\"#5F*$\"$+\"F*-%'COLOURG6&%$
RGBG$FQ!\"\"$F*F*FZ-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%V
IEWG6$;FZ$\"#7F*%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "This g
ives a nice picture of the sequence " }{XPPEDIT 18 0 "a;" "6#%\"aG" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a_n = n^2;" "6#/%$a_nG*$%\"nG\"
\"#" }{TEXT -1 43 " .  The next example is the sequence whose " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 12 "-th term is " }{XPPEDIT 18 
0 "n/(n+1);" "6#*&%\"nG\"\"\",&F$F%F%F%!\"\"" }{TEXT -1 2 " ." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "b := [[n, n/(n+1)] $n=1..10]
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG7,7$\"\"\"#F'\"\"#7$F)#F)\"
\"$7$F,#F,\"\"%7$F/#F/\"\"&7$F2#F2\"\"'7$F5#F5\"\"(7$F8#F8\"\")7$F;#F;
\"\"*7$F>#F>\"#57$FA#FA\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "plot(b, x=0..12, style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
300 300 {PLOTDATA 2 "6&-%'CURVESG6$7,7$$\"\"\"\"\"!$\"3++++++++]!#=7$$
\"\"#F*$\"3ImmmmmmmmF-7$$\"\"$F*$\"3++++++++vF-7$$\"\"%F*$\"3U+++++++!
)F-7$$\"\"&F*$\"3qLLLLLLL$)F-7$$\"\"'F*$\"3%4dG9dG9d)F-7$$\"\"(F*$\"3+
++++++]()F-7$$\"\")F*$\"3S))))))))))))))))F-7$$\"\"*F*$\"3A+++++++!*F-
7$$\"#5F*$\"3g!4444444*F--%'COLOURG6&%$RGBG$FX!\"\"$F*F*F[o-%&STYLEG6#
%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F[o$\"#7F*%(DEFAULTG" 
1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Notice that the sequences " }
{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b;" "
6#%\"bG" }{TEXT -1 36 " behave quite differently for large " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 15 ": the sequence " }{XPPEDIT 
18 0 "a;" "6#%\"aG" }{TEXT -1 28 " grows without bound, while " }
{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 70 " appears to approach a fini
te value.  We can confirm this by plotting " }{XPPEDIT 18 0 "b;" "6#%
\"bG" }{TEXT -1 20 " for more values of " }{XPPEDIT 18 0 "n;" "6#%\"nG
" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([[
n, n/(n+1)] $n=1..100], x=0..100, y=0..2, style=point);" }}{PARA 13 "
" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7`q7$$\"\"\"\"
\"!$\"3++++++++]!#=7$$\"\"#F*$\"3ImmmmmmmmF-7$$\"\"$F*$\"3++++++++vF-7
$$\"\"%F*$\"3U+++++++!)F-7$$\"\"&F*$\"3qLLLLLLL$)F-7$$\"\"'F*$\"3%4dG9
dG9d)F-7$$\"\"(F*$\"3+++++++]()F-7$$\"\")F*$\"3S))))))))))))))))F-7$$
\"\"*F*$\"3A+++++++!*F-7$$\"#5F*$\"3g!4444444*F-7$$\"#6F*$\"3Immmmmmm
\"*F-7$$\"#7F*$\"3GJ#p2Bp2B*F-7$$\"#8F*$\"3-'G9dG9dG*F-7$$\"#9F*$\"3[L
LLLLLL$*F-7$$\"#:F*$\"3+++++++v$*F-7$$\"#;F*$\"3\"GN#)eqk<T*F-7$$\"#<F
*$\"3?WWWWWWW%*F-7$$\"#=F*$\"3E:j_5Uot%*F-7$$\"#>F*$\"3a*************
\\*F-7$$\"#?F*$\"3GB&4Q_4Q_*F-7$$\"#@F*$\"3'eaaaaaaa*F-7$$\"#AF*$\"37[
VI\"R<_c*F-7$$\"#BF*$\"3qLLLLLL$e*F-7$$\"#CF*$\"3k*************f*F-7$$
\"#DF*$\"3k:YQ:YQ:'*F-7$$\"#EF*$\"35H'H'H'H'H'*F-7$$\"#FF*$\"3,Vr&G9dG
k*F-7$$\"#GF*$\"3)Q5$z8C<b'*F-7$$\"#HF*$\"3ummmmmmm'*F-7$$\"#IF*$\"3'*
4(Q[N>un*F-7$$\"#JF*$\"3++++++](o*F-7$$\"#KF*$\"3C(ppppppp*F-7$$\"#LF*
$\"3Tw6%HN#)eq*F-7$$\"#MF*$\"3>9dG9dG9(*F-7$$\"#NF*$\"35AAAAAAA(*F-7$$
\"#OF*$\"3FI(H(H(H(H(*F-7$$\"#PF*$\"3>eJE0@%ot*F-7$$\"#QF*$\"3RV(*eV(*
eV(*F-7$$\"#RF*$\"3y************\\(*F-7$$\"#SF*$\"3a4c(4c(4c(*F-7$$\"#
TF*$\"3khZ!>w/>w*F-7$$\"#UF*$\"3'e6l/'=Wn(*F-7$$\"#VF*$\"3#HFFFFFFx*F-
7$$\"#WF*$\"3Yxxxxxxx(*F-7$$\"#XF*$\"31u@l&p3Ey*F-7$$\"#YF*$\"3=\">`D/
Msy*F-7$$\"#ZF*$\"3Immmmmm\"z*F-7$$\"#[F*$\"3aQpMn$=fz*F-7$$\"#\\F*$\"
3#)*************z*F-7$$\"#]F*$\"3d]uio:#R!)*F-7$$\"#^F*$\"3F2Bp2Bp2)*F
-7$$\"#_F*$\"3<\")pra2K6)*F-7$$\"#`F*$\"35:[\"[\"[\"[\")*F-7$$\"#aF*$
\"37=======)*F-7$$\"#bF*$\"3&4dG9dG9#)*F-7$$\"#cF*$\"3Rr(3NShX#)*F-7$$
\"#dF*$\"3Q^l*o?'eF)*F-7$$\"#eF*$\"3kridu%30$)*F-7$$\"#fF*$\"3#GLLLLLL
$)*F-7$$\"#gF*$\"3i\"\\qPdlg$)*F-7$$\"#hF*$\"3)\\N>un4(Q)*F-7$$\"#iF*$
\"3tS)p7%)p7%)*F-7$$\"#jF*$\"3++++++vV)*F-7$$\"#kF*$\"3qYQ:YQ:Y)*F-7$$
\"#lF*$\"3i[[[[[[[)*F-7$$\"#mF*$\"3+;nloiu])*F-7$$\"#nF*$\"3w)eqk<TH&)
*F-7$$\"#oF*$\"3U;\"oPYs]&)*F-7$$\"#pF*$\"3mdG9dG9d)*F-7$$\"#qF*$\"3)
\\Yx&H\\:f)*F-7$$\"#rF*$\"3g666666h)*F-7$$\"#sF*$\"3WO,j)p8I')*F-7$$\"
#tF*$\"39l['['['[')*F-7$$\"#uF*$\"3#pmmmmmm')*F-7$$\"#vF*$\"3ay:j_5Uo)
*F-7$$\"#wF*$\"35q)H,()H,()*F-7$$\"#xF*$\"3Es[zr[zr)*F-7$$\"#yF*$\"3y(
)*=:s<M()*F-7$$\"#zF*$\"3W++++++v)*F-7$$\"#!)F*$\"3mUl()4Kaw)*F-7$$\"#
\")F*$\"3y/y[!y[!y)*F-7$$\"##)F*$\"3Cc\"*Gs!=&z)*F-7$$\"#$)F*$\"3Q\"Q_
4Q_4))*F-7$$\"#%)F*$\"3+rk<THN#))*F-7$$\"#&)F*$\"3%zbK-$4s$))*F-7$$\"#
')F*$\"3'zOk7Zd]))*F-7$$\"#()F*$\"3YOOOOOO'))*F-7$$\"#))F*$\"3)>?Q%\\/
k())*F-7$$\"#*)F*$\"3G*))))))))))))))*F-7$$\"#!*F*$\"3]!*)4,*)4,*)*F-7
$$\"#\"*F*$\"3['3EyM/8*)*F-7$$\"##*F*$\"3)*p&z#=JZ#*)*F-7$$\"#$*F*$\"3
9'fw7-<O*)*F-7$$\"#%*F*$\"3#GE0@%ot%*)*F-7$$\"#&*F*$\"3qLLLLL$e*)*F-7$
$\"#'*F*$\"3sX[\\;s!p*)*F-7$$\"#(*F*$\"3GpMn$=fz*)*F-7$$\"#)*F*$\"3W**
)*)*)*)*)*)*)*F-7$$\"#**F*$\"3!***************)*F-7$$\"$+\"F*$\"3!4!*4
!*4!*4!**F--%'COLOURG6&%$RGBG$FX!\"\"$F*F*F][m-%&STYLEG6#%&POINTG-%+AX
ESLABELSG6$Q\"x6\"Q\"yFf[m-%%VIEWG6$;F][mFcjl;F][mF/" 1 5 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 28 "It now seems clear that, as " }{XPPEDIT 18 0 "n;
" "6#%\"nG" }{TEXT -1 26 " becomes large, the terms " }{XPPEDIT 18 0 "
b_n = n/(n+1);" "6#/%$b_nG*&%\"nG\"\"\",&F&F'F'F'!\"\"" }{TEXT -1 60 "
 approach 1.  (This should also be clear from algebra: when " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 126 " is very large, the '+1' i
n the denominator makes very little difference, so the fraction should
 be approximately the same as " }{XPPEDIT 18 0 "n/n = 1;" "6#/*&%\"nG
\"\"\"F%!\"\"F&" }{TEXT -1 3 " .)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "In an example such as sequence " }
{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 60 ", where the general term ap
proaches a fixed, finite, number " }{XPPEDIT 18 0 "L;" "6#%\"LG" }
{TEXT -1 4 " as " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 36 " becomes
 large, we say the sequence " }{TEXT 265 9 "converges" }{TEXT -1 4 " t
o " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 11 ", or has a " }{TEXT 
266 5 "limit" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT 
-1 14 ", and we write" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "limit(b_n,n \+
= infinity) = L;" "6#/-%&limitG6$%$b_nG/%\"nG%)infinityG%\"LG" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 36 "Otherwise, we say that the se
quence " }{TEXT 267 8 "diverges" }{TEXT -1 73 ", or that the limit doe
s not exist.  For example, we have seen above that" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "limit(n/(n+1),n = infinity) = 1;" "6#/-%&limitG6$*&%\"n
G\"\"\",&F(F)F)F)!\"\"/F(%)infinityGF)" }{TEXT -1 12 " , but that " }
{XPPEDIT 18 0 "limit(n^2,n = infinity);" "6#-%&limitG6$*$%\"nG\"\"#/F'
%)infinityG" }{TEXT -1 16 " does not exist." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question \+
2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 250 "Graph the sequences whose ge
neral terms are given by the following expressions, and determine whet
her or not they converge.  If they do, try to find the limit.  (You mi
ght try to guess the answers before drawing the graphs; they are not a
ll obvious.)" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a)  " }{XPPEDIT 18 0 "(-1
)^n;" "6#),$\"\"\"!\"\"%\"nG" }{TEXT -1 21 "                 (b) " }
{XPPEDIT 18 0 "(1+1/n)^n;" "6#),&\"\"\"F%*&F%F%%\"nG!\"\"F%F'" }{TEXT 
-1 18 "              (c) " }{XPPEDIT 18 0 "(1+2/n)^n;" "6#),&\"\"\"F%*
&\"\"#F%%\"nG!\"\"F%F(" }{TEXT -1 22 "                 (d)  " }
{XPPEDIT 18 0 "(1-1/n)^n;" "6#),&\"\"\"F%*&F%F%%\"nG!\"\"F(F'" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(e) " }{XPPEDIT 18 0 "sqrt(n+1)-
sqrt(n);" "6#,&-%%sqrtG6#,&%\"nG\"\"\"F)F)F)-F%6#F(!\"\"" }{TEXT -1 
11 "       (f) " }{XPPEDIT 18 0 "sqrt(n^2+n)-n;" "6#,&-%%sqrtG6#,&*$%
\"nG\"\"#\"\"\"F)F+F+F)!\"\"" }{TEXT -1 13 "         (g) " }{XPPEDIT 
18 0 "sin(n);" "6#-%$sinG6#%\"nG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 274 10 "Solutions
." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([[
n, (-1)^n] $n=1..20], x=0..20, style=point);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$767$$\"\"\"\"\"!$!\"
\"F*7$$\"\"#F*F(7$$\"\"$F*F+7$$\"\"%F*F(7$$\"\"&F*F+7$$\"\"'F*F(7$$\"
\"(F*F+7$$\"\")F*F(7$$\"\"*F*F+7$$\"#5F*F(7$$\"#6F*F+7$$\"#7F*F(7$$\"#
8F*F+7$$\"#9F*F(7$$\"#:F*F+7$$\"#;F*F(7$$\"#<F*F+7$$\"#=F*F(7$$\"#>F*F
+7$$\"#?F*F(-%'COLOURG6&%$RGBG$FGF,$F*F*Feo-%&STYLEG6#%&POINTG-%+AXESL
ABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;FeoF^o%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 119 "Sequence (a) does not converge: the terms bounce back \+
and forth between 1 and -1, and do not approach any fixed number." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([[n, (1 + 1/n)^n] $n=1.
.30], x=0..30, style=point);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6&-%'CURVESG6$7@7$$\"\"\"\"\"!$\"\"#F*7$F+$\"3+++++++
]A!#<7$$\"\"$F*$\"3Cq.Pq.PqBF07$$\"\"%F*$\"3++++]iSTCF07$$\"\"&F*$\"3'
)*********>$)[#F07$$\"\"'F*$\"3d7@urji@DF07$$\"\"(F*$\"3+82/(p*\\YDF07
$$\"\")F*$\"3!zM]R^%ylDF07$$\"\"*F*$\"33(>8<zu6e#F07$$\"#5F*$\"33++5gC
u$f#F07$$\"#6F*$\"3*3`(*=,*>/EF07$$\"#7F*$\"31yYA!HNIh#F07$$\"#8F*$\"3
4Kd)y)3g?EF07$$\"#9F*$\"3Rp3Ic::FEF07$$\"#:F*$\"3:>zs<(yGj#F07$$\"#;F*
$\"3'**fmt\\Gzj#F07$$\"#<F*$\"324J=vVTUEF07$$\"#=F*$\"3?&o(4@eUYEF07$$
\"#>F*$\"3qW/kEV.]EF07$$\"#?F*$\"3U?W90xH`EF07$$\"#@F*$\"3K0h#R@jil#F0
7$$\"#AF*$\"3M)yP&e)p*eEF07$$\"#BF*$\"3y\"yQ'=,XhEF07$$\"#CF*$\"3M%fo!
e7tjEF07$$\"#DF*$\"3#)>u[Jj$em#F07$$\"#EF*$\"3ISP`m\\ynEF07$$\"#FF*$\"
3MuD\"y(RfpEF07$$\"#GF*$\"3oR3W`yFrEF07$$\"#HF*$\"3I,3)R9\\Gn#F07$$\"#
IF*$\"3k%Hqex=Vn#F0-%'COLOURG6&%$RGBG$FV!\"\"$F*F*F]u-%&STYLEG6#%&POIN
TG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F]uFct%(DEFAULTG" 1 5 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "Sequence (b) appears to converge to a li
mit which is approximately 2.7.  (Actually, this sequence converges to
 the number " }{XPPEDIT 18 0 "e;" "6#%\"eG" }{TEXT -1 48 ", which is a
pproximately equal to 2.7182818285.)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "plot([[n, (1 + 2/n)^n] $n=1..30], x=0..30, style=poin
t);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVES
G6$7@7$$\"\"\"\"\"!$\"\"$F*7$$\"\"#F*$\"\"%F*7$F+$\"3wH'H'H'H'HY!#<7$F
0$\"3++++++]i]F57$$\"\"&F*$\"3!**********R#y`F57$$\"\"'F*$\"3Rm(HFpb'=
cF57$$\"\"(F*$\"3%*yA&**4&z2eF57$$\"\")F*$\"3]i!RvZk/'fF57$$\"\"*F*$\"
3YRv=S^F'3'F57$$\"#5F*$\"3)*****RAkt\">'F57$$\"#6F*$\"3=q8-]GT\"G'F57$
$\"#7F*$\"3!3Hl'e&*fejF57$$\"#8F*$\"3v6cJaYtDkF57$$\"#9F*$\"3mV%GqqgY[
'F57$$\"#:F*$\"3`(f'oYgzOlF57$$\"#;F*$\"3-Bu-s,D$e'F57$$\"#<F*$\"30*zQ
8W.\\i'F57$$\"#=F*$\"3!fmv`IjCm'F57$$\"#>F*$\"3o>iXfZ]'p'F57$$\"#?F*$
\"3D+cK\\**\\FnF57$$\"#@F*$\"3;c*Hx<Sev'F57$$\"#AF*$\"3'f2oN\\_=y'F57$
$\"#BF*$\"3N^h*yN7e!oF57$$\"#CF*$\"3<o&fzU`z#oF57$$\"#DF*$\"3+;$>i>v%[
oF57$$\"#EF*$\"3#z[(e8!\\v'oF57$$\"#FF*$\"3?\\!o>)GK&)oF57$$\"#GF*$\"3
a!3u(*HD>!pF57$$\"#HF*$\"3[)4e<Pou\"pF57$$\"#IF*$\"3f6YEU.0KpF5-%'COLO
URG6&%$RGBG$FT!\"\"$F*F*F[u-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q
!6\"-%%VIEWG6$;F[uFat%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
94 "This sequence appears to converge to a limit of approximately 7.  \+
(The exact limit is in fact " }{XPPEDIT 18 0 "e^2;" "6#*$%\"eG\"\"#" }
{TEXT -1 2 ".)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot([[n,
 (1 - 1/n)^n] $n=1..30], x=0..30, style=point);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7@7$$\"\"\"\"\"!$F*F*
7$$\"\"#F*$\"3++++++++D!#=7$$\"\"$F*$\"3!G'H'H'H'H'HF17$$\"\"%F*$\"3++
+++D1kJF17$$\"\"&F*$\"3G+++++!oF$F17$$\"\"'F*$\"3oSQ!owz*[LF17$$\"\"(F
*$\"3;P6*3xm\"*R$F17$$\"\")F*$\"3]m\"e!e\"*3OMF17$$\"\"*F*$\"3O&=Y6;%R
kMF17$$\"#5F*$\"3')*****4S%y'[$F17$$\"#6F*$\"3zCR\"[**Q\\]$F17$$\"#7F*
$\"3&3PT,Gc*>NF17$$\"#8F*$\"3(eg6Z)\\eKNF17$$\"#9F*$\"3n)e)>-JNVNF17$$
\"#:F*$\"3oT9%\\mVEb$F17$$\"#;F*$\"39Gz^/8ugNF17$$\"#<F*$\"31GHYZ>'yc$
F17$$\"#=F*$\"3GNm%zOsTd$F17$$\"#>F*$\"3'ejM?U.)zNF17$$\"#?F*$\"3oAa3C
#f[e$F17$$\"#@F*$\"3#3`4kkB%*e$F17$$\"#AF*$\"3sgg&4,lNf$F17$$\"#BF*$\"
3Y<9!Q`Rtf$F17$$\"#CF*$\"3+B_&G*Qz+OF17$$\"#DF*$\"3t#=!eor'Rg$F17$$\"#
EF*$\"3nS]OHB*og$F17$$\"#FF*$\"3*e84:Q(f4OF17$$\"#GF*$\"3!3Uo*oi57OF17
$$\"#HF*$\"3[\\*pTeRWh$F17$$\"#IF*$\"3=0hhM^h;OF1-%'COLOURG6&%$RGBG$FW
!\"\"F+F+-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F+F
dt%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 
"Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Sequence (d) again a
ppears to converge.  (This time, the exact limit is " }{XPPEDIT 18 0 "
e^(-1);" "6#)%\"eG,$\"\"\"!\"\"" }{TEXT -1 33 "; do you begin to see a
 pattern?)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([[n, sqr
t(n+1) - sqrt(n)] $n=1..30], x=0..30, style=point);" }}{PARA 13 "" 1 "
" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7@7$$\"\"\"\"\"!$\"
3Y^4tBc8UT!#=7$$\"\"#F*$\"3[?y&>Xs$yJF-7$$\"\"$F*$\"31G7JC>\\zEF-7$$\"
\"%F*$\"31)*y*\\xz1O#F-7$$\"\"&F*$\"3w!)Q$Gl<U8#F-7$$\"\"'F*$\"3OGT\"G
o:E'>F-7$$\"\"(F*$\"3v&*f\"o8en#=F-7$$\"\")F*$\"34(4QDvGdr\"F-7$$\"\"*
F*$\"3B&z$o,mxA;F-7$$\"#5F*$\"3(G?q=IrMa\"F-7$$\"#6F*$\"3wXN#yCoZZ\"F-
7$$\"#7F*$\"3QZBE.m\\99F-7$$\"#8F*$\"3/A&*4861h8F-7$$\"#9F*$\"3#pvMVff
KJ\"F-7$$\"#:F*$\"3yHe#z`m,F\"F-7$$\"#;F*$\"3'egwhDc5B\"F-7$$\"#<F*$\"
3%=C;]h]`>\"F-7$$\"#=F*$\"31#*Q@kDei6F-7$$\"#>F*$\"3Mc!*e9,PK6F-7$$\"#
?F*$\"3>-Ec*R(R/6F-7$$\"#@F*$\"3-**en[1Sy5F-7$$\"#AF*$\"3#Q*G*[jdT0\"F
-7$$\"#BF*$\"3\\mj`A'z9.\"F-7$$\"#CF*$\"3PUkLW^?55F-7$$\"#DF*$\"3;\\%y
#f8&>!**!#>7$$\"#EF*$\"3?`ZQ64H8(*F`s7$$\"#FF*$\"3(3%\\DU*>]`*F`s7$$\"
#GF*$\"3sIA`+&=iO*F`s7$$\"#HF*$\"3*[u:<zwg?*F`s7$$\"#IF*$\"3-Lg$yxyQ0*
F`s-%'COLOURG6&%$RGBG$FX!\"\"$F*F*F`u-%&STYLEG6#%&POINTG-%+AXESLABELSG
6$Q\"x6\"Q!6\"-%%VIEWG6$;F`uFft%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 148 "Sequence (e) appears to converge to 0.  This is an inter
esting example: the general term in the sequence is the difference of \+
two large numbers (if " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 38 " i
s large), but it is not the numbers " }{XPPEDIT 18 0 "sqrt(n+1);" "6#-
%%sqrtG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "sqrt(n)
;" "6#-%%sqrtG6#%\"nG" }{TEXT -1 178 " individually that count, but th
eir difference; and the difference does indeed go to 0.  The next exam
ple shows that it is not always obvious when such a difference will go
 to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([[n, sqrt(n^
2 + n) - n] $n=1..30], x=0..30, style=point);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7@7$$\"\"\"\"\"!$\"3Y
^4tBc8UT!#=7$$\"\"#F*$\"3!)y<$yU(*[\\%F-7$$\"\"$F*$\"3'Qax8:;5k%F-7$$
\"\"%F*$\"35'z&**\\&f8s%F-7$$\"\"&F*$\"3)=h;0vbAx%F-7$$\"\"'F*$\"3y.'y
S)pS2[F-7$$\"\"(F*$\"3fE)yatZJ$[F-7$$\"\")F*$\"3U&p&QUP\"G&[F-7$$\"\"*
F*$\"3C\"Q^]!)H$o[F-7$$\"#5F*$\"3?a^,<[)3)[F-7$$\"#6F*$\"3Ks0wIHD\"*[F
-7$$\"#7F*$\"3=mz'z'*f***[F-7$$\"#8F*$\"3^;/KKcP2\\F-7$$\"#9F*$\"3_!Q%
*=YnP\"\\F-7$$\"#:F*$\"3'3o'H[QL>\\F-7$$\"#;F*$\"3VBkqC]AC\\F-7$$\"#<F
*$\"3^/!f`%obG\\F-7$$\"#=F*$\"3J*Gp!*3?C$\\F-7$$\"#>F*$\"3Ni#zh*o)e$\\
F-7$$\"#?F*$\"3Zf>>>`,R\\F-7$$\"#@F*$\"3Nsn/-E&=%\\F-7$$\"#AF*$\"3c`)R
SePW%\\F-7$$\"#BF*$\"3A,YT*[-o%\\F-7$$\"#CF*$\"3!f!yJyU(*[\\F-7$$\"#DF
*$\"3MU#R'zc(4&\\F-7$$\"#EF*$\"3e'RN)*)f#G&\\F-7$$\"#FF*$\"3]2/N(pTX&
\\F-7$$\"#GF*$\"3WH,]vp8c\\F-7$$\"#HF*$\"3C+D0vSid\\F-7$$\"#IF*$\"3!z7
Q&RO,f\\F--%'COLOURG6&%$RGBG$FX!\"\"$F*F*F_u-%&STYLEG6#%&POINTG-%+AXES
LABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F_uFet%(DEFAULTG" 1 5 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 140 "In this case, we again have the difference of two l
arge numbers.  A reasonable approach to understanding this limit would
 be to say that if " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 16 " is l
arge, then " }{XPPEDIT 18 0 "n^2;" "6#*$%\"nG\"\"#" }{TEXT -1 21 " is \+
much bigger than " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 46 ", and s
o we should be able to ignore the term " }{XPPEDIT 18 0 "n;" "6#%\"nG
" }{TEXT -1 74 " under the square root.  This would mean that the gene
ral term (for large " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 19 ") sh
ould look like " }{XPPEDIT 18 0 "sqrt(n^2)-n = 0;" "6#/,&-%%sqrtG6#*$%
\"nG\"\"#\"\"\"F)!\"\"\"\"!" }{TEXT -1 3 ".  " }{TEXT 275 0 "" }{TEXT 
-1 31 "As you can see from the graph, " }{TEXT 276 23 "this argument i
s wrong!" }{TEXT -1 128 "  In fact, this sequence converges to 1/2.  (
The explanation for this fact is in Section 10.11 of Stewart: the Bino
mial Series.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([[n, \+
sin(n)] $n=1..30], x=0..30, style=point);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7@7$$\"\"\"\"\"!$\"30
l*y![)4ZT)!#=7$$\"\"#F*$\"35<oDoU(H4*F-7$$\"\"$F*$\"38s')f!3+7T\"F-7$$
\"\"%F*$!3-#GzI&\\-ovF-7$$\"\"&F*$!3a%QJmuU#*e*F-7$$\"\"'F*$!3ge#*)>)
\\:%z#F-7$$\"\"(F*$\"3i!*y=()f')plF-7$$\"\")F*$\"3w;QBmCe$*)*F-7$$\"\"
*F*$\"3%fc<C&[=@TF-7$$\"#5F*$!3&))p$*)36@SaF-7$$\"#6F*$!3uMq]l?!*****F
-7$$\"#7F*$!3S\\V+!=HdO&F-7$$\"#8F*$\"3;4kEo.n,UF-7$$\"#9F*$\"3S-([pbt
g!**F-7$$\"#:F*$\"3Op6d,%yG]'F-7$$\"#;F*$!3+`1lmJ.zGF-7$$\"#<F*$!3?pbz
=\\(Rh*F-7$$\"#=F*$!3agnrnC()4vF-7$$\"#>F*$\"3UB&Hm4s()\\\"F-7$$\"#?F*
$\"3mwiF2DXH\"*F-7$$\"#@F*$\"3og0O&QclO)F-7$$\"#AF*$!3AwQS!H48&))!#?7$
$\"#BF*$!3Q1<vTS?i%)F-7$$\"#CF*$!3%*Qi1?Oyb!*F-7$$\"#DF*$!3IIx(4]<NK\"
F-7$$\"#EF*$\"3,Ggz/XeDwF-7$$\"#FF*$\"3GI]/%GfPc*F-7$$\"#GF*$\"3W!pyI)
y04FF-7$$\"#HF*$!3Sv'H@%)Qjj'F-7$$\"#IF*$!3E='G4C;.))*F--%'COLOURG6&%$
RGBG$FX!\"\"$F*F*F`u-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%
VIEWG6$;F`uFft%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 324 "This \+
sequence is actually the hardest of all to understand rigorously, but \+
it certainly looks from the graph as if it does not converge.  (This i
s in fact the right answer.)  There doesn't seem to be any pattern dev
eloping, however, that makes the divergence obvious.  We could try plo
tting a few more terms of the sequence:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "plot([[n, sin(n)] $n=1..100], x=0..100, style=point);
" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$
7`q7$$\"\"\"\"\"!$\"30l*y![)4ZT)!#=7$$\"\"#F*$\"35<oDoU(H4*F-7$$\"\"$F
*$\"38s')f!3+7T\"F-7$$\"\"%F*$!3-#GzI&\\-ovF-7$$\"\"&F*$!3a%QJmuU#*e*F
-7$$\"\"'F*$!3ge#*)>)\\:%z#F-7$$\"\"(F*$\"3i!*y=()f')plF-7$$\"\")F*$\"
3w;QBmCe$*)*F-7$$\"\"*F*$\"3%fc<C&[=@TF-7$$\"#5F*$!3&))p$*)36@SaF-7$$
\"#6F*$!3uMq]l?!*****F-7$$\"#7F*$!3S\\V+!=HdO&F-7$$\"#8F*$\"3;4kEo.n,U
F-7$$\"#9F*$\"3S-([pbtg!**F-7$$\"#:F*$\"3Op6d,%yG]'F-7$$\"#;F*$!3+`1lm
J.zGF-7$$\"#<F*$!3?pbz=\\(Rh*F-7$$\"#=F*$!3agnrnC()4vF-7$$\"#>F*$\"3UB
&Hm4s()\\\"F-7$$\"#?F*$\"3mwiF2DXH\"*F-7$$\"#@F*$\"3og0O&QclO)F-7$$\"#
AF*$!3AwQS!H48&))!#?7$$\"#BF*$!3Q1<vTS?i%)F-7$$\"#CF*$!3%*Qi1?Oyb!*F-7
$$\"#DF*$!3IIx(4]<NK\"F-7$$\"#EF*$\"3,Ggz/XeDwF-7$$\"#FF*$\"3GI]/%GfPc
*F-7$$\"#GF*$\"3W!pyI)y04FF-7$$\"#HF*$!3Sv'H@%)Qjj'F-7$$\"#IF*$!3E='G4
C;.))*F-7$$\"#JF*$!3?]1B`kPSSF-7$$\"#KF*$\"3'e!pT7oE9bF-7$$\"#LF*$\"3!
=ns5g=\"****F-7$$\"#MF*$\"3[Q-?ho#3H&F-7$$\"#NF*$!3C5:'\\pE=G%F-7$$\"#
OF*$!3#y:JW`)y<**F-7$$\"#PF*$!3(Q**pNL\"QNkF-7$$\"#QF*$\"3A`Q4(y&ojHF-
7$$\"#RF*$\"3&y(3%G'Q&zj*F-7$$\"#SF*$\"3O)[$z/;8^uF-7$$\"#TF*$!3))*3Z!
)oEie\"F-7$$\"#UF*$!3%yLc\"za@l\"*F-7$$\"#VF*$!3'H)fGEuu<$)F-7$$\"#WF*
$\"3wd8a5D>q<!#>7$$\"#XF*$\"3Q%=T`CN!4&)F-7$$\"#YF*$\"3>#4)[wM)y,*F-7$
$\"#ZF*$\"3,SAXF7tN7F-7$$\"#[F*$!3?omB8ma#o(F-7$$\"#\\F*$!3`=ZfFl_P&*F
-7$$\"#]F*$!3n(GRq`[Pi#F-7$$\"#^F*$\"39ZPVe<H-nF-7$$\"#_F*$\"3C_[S?fFm
)*F-7$$\"#`F*$\"3=U$==]^#fRF-7$$\"#aF*$!3yhh^)[!*ye&F-7$$\"#bF*$!3_)>'
eL<b(***F-7$$\"#cF*$!3_=\"p3-5b@&F-7$$\"#dF*$\"3U\\#yCbZ;O%F-7$$\"#eF*
$\"3Cr`%3[E(G**F-7$$\"#fF*$\"3iy8Rr+QnjF-7$$\"#gF*$!3%o;A5@1\"[IF-7$$
\"#hF*$!3SHR3+x<h'*F-7$$\"#iF*$!3CHA\\mp!=R(F-7$$\"#jF*$\"39p!GI+dNn\"
F-7$$\"#kF*$\"3I1z'>Qg-?*F-7$$\"#lF*$\"3CN5!\\z'Go#)F-7$$\"#mF*$!3WzmR
-a6bEF`y7$$\"#nF*$!3#GA`(*y*>b&)F-7$$\"#oF*$!3)H\"H*o!oFz*)F-7$$\"#pF*
$!3Cs=$y8[y9\"F-7$$\"#qF*$\"3$4*)yb\"o!*QxF-7$$\"#rF*$\"3gYPaKla5&*F-7
$$\"#sF*$\"3ki.iFOBQDF-7$$\"#tF*$!3CwI()o&>xw'F-7$$\"#uF*$!3#QZ#o/EY^)
*F-7$$\"#vF*$!3[/V4aj\"y(QF-7$$\"#wF*$\"3K.=)*oj2hcF-7$$\"#xF*$\"3w6t!
ee,_***F-7$$\"#yF*$\"3@_`()fXyR^F-7$$\"#zF*$!3u$3vqoE6W%F-7$$\"#!)F*$!
35^PBRl))Q**F-7$$\"#\")F*$!3kRXuU*z))H'F-7$$\"##)F*$\"3i^3LCyGKJF-7$$
\"#$)F*$\"3#[&=+6Yk$o*F-7$$\"#%)F*$\"3,@Ht+K!>L(F-7$$\"#&)F*$!3'3(e[*>
c2w\"F-7$$\"#')F*$!3o(fS+Z%eM#*F-7$$\"#()F*$!3HD#3jOy\"=#)F-7$$\"#))F*
$\"3togOt-$)RNF`y7$$\"#*)F*$\"3!G`C\"eSp+')F-7$$\"#!*F*$\"3]yb+Om'*R*)
F-7$$\"#\"*F*$\"3_o:^<^()f5F-7$$\"##*F*$!3oY!ehpgYz(F-7$$\"#$*F*$!3as%
*p79#G[*F-7$$\"#%*F*$!35Vlna)>DX#F-7$$\"#&*F*$\"3i47OZrhKoF-7$$\"#'*F*
$\"32\\MMau(e$)*F-7$$\"#(*F*$\"3];_F!Rxgz$F-7$$\"#)*F*$!3)*GU!*>(=Qt&F
-7$$\"#**F*$!3)p`j=Mo?***F-7$$\"$+\"F*$!3!ze(46klj]F--%'COLOURG6&%$RGB
G$FX!\"\"$F*F*F_[m-%&STYLEG6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VI
EWG6$;F_[mFejl%(DEFAULTG" 1 5 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 289 "There
 are more points plotted, but still nothing obvious that will confirm \+
the divergence.  In fact, this sequence will never settle down to any \+
\"simple\" behaviour, such as bouncing between two values: no matter h
ow many points you plot, the picture will look somewhat like the ones \+
above." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Of course, " }{TEXT 
268 5 "Maple" }{TEXT -1 34 " can compute limits algebraically:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "limit(n/(n+1), n=infinity);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 3
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Use the " }{TEXT 269 5 "limit" 
}{TEXT -1 141 " command to verify your results from Question 2.  Use i
t on each sequence, even the ones you thought were divergent, so that \+
you can see how " }{TEXT 270 5 "Maple" }{TEXT -1 40 " indicates that t
he limits do not exist." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 277 10 "Solutions." }{TEXT -1 11 "  Here are
 " }{TEXT 278 7 "Maple's" }{TEXT -1 44 " computations of the seven lim
its, in order." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit((-1
)^n, n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#;!\"\"\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "We saw in Question 2 that this se
quence bounces back and forth between 1 and -1, so it has no limit.  \+
" }{TEXT 279 5 "Maple" }{TEXT -1 56 " gives the \"limit\" as a range: \+
between 1 and -1.  Since " }{TEXT 280 5 "Maple" }{TEXT -1 134 " has no
t given the limit as a single number, you should conclude from this ou
tput that the limit, as defined in class, does not exist." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((1 + 1/n)^n, n=infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 16 "It appears that " }{TEXT 281 5 "Maple" }{TEXT -1 63 
" is aware of the relation between this sequence and the number " }
{XPPEDIT 18 0 "e;" "6#%\"eG" }{TEXT -1 35 ".  What about the other var
iations?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((1 + 2/n)
^n, n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"#" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((1 - 1/n)^n, n=infini
ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 54 "Hmmm...what about the general pattern?  (
Just in case " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 75 " was assign
ed a value elsewhere in the worksheet, let's unassign it first.)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x := 'x';" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"xGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "limit((1 + x/n)^n, n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$expG6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Interesting!  \+
Apparently, the exponential function can be computed as the limit of a
 sequence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "limit(sqrt(n+
1) - sqrt(n), n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "limit(sqrt(n^2 + n) - n, n
=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "These two limits are consistent wi
th what we saw in Question 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "limit(sin(n), n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#;!
\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "As in the first seq
uence, " }{TEXT 282 5 "Maple" }{TEXT -1 61 " indicates this sequence d
oes not converge by giving a range." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 8 "Warning:" }{TEXT 
-1 128 "  In this worksheet, we have not discussed the rigorous defini
tion of limit, which is written in a box on page 580 of Stewart.  " }
{TEXT 272 45 "You will be expected to know this definition," }{TEXT 
-1 59 " and to be able to use it to show that simple limits exist." }}
}}}{MARK "1 16 12 1 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }