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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Module 6 : Precalculus" }}{PARA 
3 "" 0 "" {TEXT -1 24 "605 : Sequences & Series" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" 
{TEXT -1 252 "In this project we will use the following command packag
es. Type and execute this line before begining the project below. If y
ou re-enter the worksheet for this project, be sure to re-execute this
 statement before jumping to any point in the worksheet." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "resta
rt; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha
ngecoords has been redefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 83 "_______________________________________________
____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 12 "A
. Sequences" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________
_________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 72 "Using the $ operator, we can easily create sequence of \+
numbers in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 54 "Prototype for the sequence, $ operator, a range for k." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "2*k^2 + 11 $ k = 1..10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"#8
\"#>\"#H\"#V\"#h\"#$)\"$4\"\"$R\"\"$t\"\"$6#" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Using this method, we can
 create many types of sequences." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "arithmetic sequences" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "2*j + 1 $ j = 1..16;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "62\"\"$\"\"&\"\"(\"\"*\"#6\"#8\"#:\"#<\"#>\"#@\"#B\"#D\"#
F\"#H\"#J\"#L" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "5*k + 17 $
 k = 1..16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"#A\"#F\"#K\"#P\"#U\"#
Z\"#_\"#d\"#i\"#n\"#s\"#x\"##)\"#()\"##*\"#(*" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "geometric sequences" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "3^k $ k = 1..16;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "62\"\"$\"\"*\"#F\"#\")\"$V#\"$H(\"%(=#\"%hl\"&
$o>\"&\\!f\"'Zr<\"'T9`\"(BVf\"\"(pHy%\")2*[V\"\")@n/V" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(4/5)^k $ k = 1..16;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "62#\"\"%\"\"&#\"#;\"#D#\"#k\"$D\"#\"$c#\"$D'#\"%C5
\"%DJ#\"%'4%\"&Dc\"#\"&%Q;\"&D\"y#\"&Ob'\"'D1R#\"'W@E\"(DJ&>#\"(w&[5\"
(Dcw*#\"(/V>%\")D\"G)[#\");sx;\"*D19W##\")k)3r'\"+DJq?7#\"*caVo#\"+Dc^
.h#\"+C=ut5\",D\"yv^I#\"+'Hn\\H%\"-D1*ye_\"" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "polynomial sequences" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k^2 + 5*k - 3 $ k = 1..16;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"$\"#6\"#@\"#L\"#Z\"#j\"#\")\"$,\"
\"$B\"\"$Z\"\"$t\"\"$,#\"$J#\"$j#\"$(H\"$L$" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "trigonometric sequences" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sin(k*Pi/2) $ k = 1..16;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"\"\"\"!!\"\"F$F#F$F%F$F#F$F%F$F#F$
F%F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "
other sequences" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "1 + (-1)^
k $ k = 1..16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "62\"\"!\"\"#F#F$F#F$F#
F$F#F$F#F$F#F$F#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "(k! +
 1) / (K^3 + k) $ k = 1..16;" }}{PARA 12 "" 1 "" {XPPMATH 20 "62,$*&\"
\"\"F%,&*$)%\"KG\"\"$F%F%F%F%!\"\"\"\"#,$*&F%F%,&F'F%F,F%F+F*,$*&F%F%,
&F'F%F*F%F+\"\"(,$*&F%F%,&F'F%\"\"%F%F+\"#D,$*&F%F%,&F'F%\"\"&F%F+\"$@
\",$*&F%F%,&F'F%\"\"'F%F+\"$@(,$*&F%F%,&F'F%F3F%F+\"%T],$*&F%F%,&F'F%
\"\")F%F+\"&@.%,$*&F%F%,&F'F%\"\"*F%F+\"'\")GO,$*&F%F%,&F'F%\"#5F%F+\"
(,)GO,$*&F%F%,&F'F%\"#6F%F+\"),o\"*R,$*&F%F%,&F'F%\"#7F%F+\"*,;+z%,$*&
F%F%,&F'F%\"#8F%F+\"+,3-Fi,$*&F%F%,&F'F%\"#9F%F+\",,7Hyr),$*&F%F%,&F'F
%\"#:F%F+\".,!oVn28,$*&F%F%,&F'F%\"#;F%F+\"/,!)))*yA4#" }}}{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 83 "_______________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 26 "B. Series & Sigma
 Notation" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________
_______________________________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 352 "When we add up a sequence of numbers the result is a sum
 or series. To express a sum in Maple, we can use the Sum command (wit
h a capital S). This command write the sum in sigma notation, but not \+
compute its value. There are two different ways of computing its value
 using the sum (with a lower case s), and the value command immediatel
y after the sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 45 "Sum( 3*k + 7, k = 1..n) = sum( k , k = 1..n);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&%\"kG\"\"$\"\"(\"\"\"/
F(;F+%\"nG,(*$),&F.F+F+F+\"\"#F+#F+F3*&#F+F3F+F.F+!\"\"#F+F3F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum( 3*k + 7, k = 1..200); v
alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,&%\"kG\"\"$\"\"(
\"\"\"/F';F*\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&+<'" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________
_______________________________________________________________" }}
{PARA 4 "" 0 "" {TEXT -1 18 "C. Rules of Series" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 222 "The distributive
 property for sums looks like this . We can verify that this rule is v
alid by computing the left and right sides of this equation, and then \+
see that the results are the same. Let k = 4k + 9, and let c = 13." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
100 "Sum( 13* (4*k + 9), k = 1..200);   Left := value(%);\nSum( 4*k + \+
9, k = 1..200);   Right := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%$SumG6$,&%\"kG\"#_\"$<\"\"\"\"/F';F*\"$+#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%LeftG\"(+'o5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$S
umG6$,&%\"kG\"\"%\"\"*\"\"\"/F';F*\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&RightG\"&+A)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 147 "We can see for ourselves that the left and right si
des are equal value, but we can ask Maple to verify that the left and \+
right values are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "testeq( Left = Right);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 83 "_______________________________________________
____________________________________" }}{PARA 4 "" 0 "" {TEXT -1 18 "D
. Series Formulas" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________
_______________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 296 "There are formulas to compute the sum of conse
cutive integers, squares, cubes, etc. You can find many of these formu
lae in your textbook or a reference book, or let Maple find the formul
a for you. To get an attractive formula, we will compute the sum, simp
lify the result, and factor that result." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "You might know this formula alread
y. This is the sum of integers : 1 + 2 + 3 + ... + n" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Sum( k, k \+
= 1..n);   value(%);   simplify(%);   factor(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$SumG6$%\"kG/F&;\"\"\"%\"nG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*$),&%\"nG\"\"\"F(F(\"\"#F(#F(F)*&#F(F)F(F'F(!\"\"#F(
F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"nG\"\"#\"\"\"#F(F'*&F)
F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"nG\"\"\",&F%F&F&F&F
&#F&\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 89 "This one is more obscure, the sum of 8th powers of integers : 1
^8 + 2^8 + 3^8 + ... + n^8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Sum( k^8, k = 1..n); value(%);   si
mplify(%);   factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*$
)%\"kG\"\")\"\"\"/F(;F*%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$),
&%\"nG\"\"\"F(F(\"\"*F(#F(F)*&#F(\"\"#F(*$)F&\"\")F(F(!\"\"*&#F-\"\"$F
()F&\"\"(F(F(*&#F6\"#:F(*$)F&\"\"&F(F(F1*&#F-F)F()F&F4F(F(*&#F(\"#IF(F
'F(F1#F(FBF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"nG\"\"$\"\"\"#
\"\"#\"\"**&#\"\"(\"#:F(*$)F&\"\"&F(F(!\"\"*&#F*F'F()F&F.F(F(*&#F(F*F(
)F&\"\")F(F(*&#F(F+F()F&F+F(F(*&#F(\"#IF(F&F(F3" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$**%\"nG\"\"\",&F%\"\"#F&F&F&,&F%F&F&F&F&,0*$)F%\"\"'F
&\"\"&*&\"#:F&)F%F.F&F&*&F.F&)F%\"\"%F&F&*&F0F&)F%\"\"$F&!\"\"*$)F%F(F
&F8*&\"\"*F&F%F&F&F7F8F&#F&\"#!*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________
____________________________________________________________________" 
}}{PARA 4 "" 0 "" {TEXT -1 45 "E. Infinite Series - Convergence & Dive
rgence" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________
____________________________________________________" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 94 "When you add up an infinite number of numbers, it is very
 like to get infinity as the result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum( 3*k - 4, k = 1..inf
inity);   % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,&
%\"kG\"\"$\"\"%!\"\"/F';\"\"\"%)infinityG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$,&%\"kG\"\"$\"\"%!\"\"/F(;\"\"\"%)infinityGF/
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "How
ever, the result is not necessarily infinite. If the numbers get small
 quickly enough, the sum may be a finite number." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum( (4/5)^k
, k = 1..infinity);   % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$SumG6$)#\"\"%\"\"&%\"kG/F*;\"\"\"%)infinityG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$)#\"\"%\"\"&%\"kG/F+;\"\"\"%)infinityGF)" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________
____________________________________________________________________" 
}}{PARA 4 "" 0 "" {TEXT -1 14 "F. Double Sums" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 272 "A more complicat
ed situation is double sum where there is a sum within a sum : . Notic
e that the limit of the inner sum, m, is the index of the outer sum. T
he inner sum is adding one number, then two numbers, then three, etc. \+
The outer sum is adding all of these together." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Although this is much mo
re complicated, we can get a formula for this too. We simply nest one \+
Sum command within another." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sum( Sum(k^2, k = 1..m), m = 1..N);
   factor( simplify( value(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
$SumG6$-F$6$*$)%\"kG\"\"#\"\"\"/F*;F,%\"mG/F/;F,%\"NG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*(%\"NG\"\"\",&F%F&\"\"#F&F&),&F%F&F&F&F(F&#F&\"
#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs( N = 100, %);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"(]3n)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }