{VERSION 5 0 "IBM INTEL NT" "5.0" }
{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 "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 
128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 
0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 
0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 128 1 0 0 0 
0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 128 1 0 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 
128 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 128 128 1 0 
0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 
0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 128 128 1 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 273 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 274 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 
1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 
0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 
}{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
279 "" 0 1 0 0 0 0 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 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Normal" -1 256 
1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 
0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 54 "High School Modul
es > Precalculus by Gregory A. Moore\n" }}{PARA 3 "" 0 "" {TEXT -1 4 "
    " }{TEXT 256 31 "Series - Finite & Infinite Sums" }}{PARA 0 "" 0 "
" {TEXT -1 86 "\nAn exploration of finite and infinite series. Recomme
nded : review sequence module .\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[D
irections : Execute the Code Resource section first. Although there wi
ll be no output immediately, these definitions are used later in this \+
worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "fade := proc( a1,a2,a3,k,n)\nlocal c;\nc := COLOR(RGB
, a1*k/n, a2*k/n, a2*k/n); \nreturn c;\nend:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 150 "FadePlot := proc( f)\n   display( [seq( plot( k
*f(x), x = 1..20, filled = true, \n             color = fade(.9,.6,.7,
k,10)), k = 0..10 )] );\nend proc:\n\n" }}}{EXCHG }}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 1 " " }{TEXT 262 26 "1. Series & Sigma Notation" }}
{PARA 0 "" 0 "" {TEXT -1 33 "\nSequences are lists of numbers.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "3*k -2  $  k = 1..20;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "((-2)^k)/ (k^2 + 1) $ k = 1.
.10;" }}}{PARA 0 "" 0 "" {TEXT -1 112 "\nWhen we add up a sequence of \+
numbers, the result is a sum or series. To express a sum in Maple, we \+
can use the " }{TEXT 277 3 "Sum" }{TEXT -1 171 " command (with a capit
al S). This command writes the sum in sigma notation, but does not com
pute its value. There are two different ways of computing its value: u
sing the " }{TEXT 278 3 "sum" }{TEXT -1 32 " (with a lower case s), an
d the " }{TEXT 279 5 "value" }{TEXT -1 31 " command immediately after \+
the " }{TEXT 276 3 "sum" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sum( 3*k -2,
 k = 1..20): % = value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "Sum( 1/k^2, k = 1..20): % = value(%);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "Sum( (3*k)/2^k, k = 1..20): % = value(%);" }}}
{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
1 " " }{TEXT 263 17 "2. Special Series" }}{PARA 0 "" 0 "" {TEXT -1 
106 "\nThere a number of \"famous\" series. It's good to be familiar w
ith these. Some are more famous, some less.\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "`Arithmetic Series`; Sum( a*k + b, k = 1..N);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`Geometric Series`; Sum( a
*r^k, k = 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "`Power \+
Series`;      Sum( 1/z^k, k = 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "`Alternating Series`; Sum( (-1)^k, k = 1..N);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "`Harmonic Series`; Sum( 1/k,
 k = 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Sum( 1/k^2, \+
k = 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Sum( 1/k!, k \+
= 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Sum(((-1)^(k+1)
)/(2*k)!, k = 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Sum
( 1/n - 1/(n+1), k = 1..N);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 18 "3. Rules of Seri
es" }}{PARA 0 "" 0 "" {TEXT -1 32 "\nHere are two rules of series.\n\n
" }{TEXT 270 17 "................ " }{TEXT 269 25 "The Distributive Pr
operty" }{TEXT 271 98 " ..............................................
..................................................." }{TEXT -1 54 "\nT
he distributive property for sums looks like this:  " }{XPPEDIT 18 0 "
Sum( c*a[k], k = 1..n)" "6#-%$SumG6$*&%\"cG\"\"\"&%\"aG6#%\"kGF(/F,;F(
%\"nG" }{TEXT -1 7 "   =   " }{XPPEDIT 18 0 "c * Sum(  a[k], k = 1..n)
" "6#*&%\"cG\"\"\"-%$SumG6$&%\"aG6#%\"kG/F,;F%%\"nGF%" }{TEXT -1 2 ".
\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "DistProp :=   Sum( c* a[k] , k = 1..N) = c*Sum(  a[k]
 , k = 1..N);" }}}{PARA 0 "" 0 "" {TEXT -1 143 "\nWe can verify that t
his rule is valid by computing the left and right sides of this equati
on, and then see that the results are the same. Let " }{XPPEDIT 18 0 "
a[[k]]" "6#&%\"aG6#7#%\"kG" }{TEXT -1 12 "  = 4k + 9.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "subs( \{a[k] = 4*k+9, c = 13, N = 5
6\}, DistProp );\nevalf(%);\ntesteq( Left = Right);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "We can see for ourselv
es that the left and right sides are equal value, but we can ask Maple
 to verify that the left and right values are the same. Here are some \+
further examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 89 "subs( \{a[k] = 1/(k^2 + 2), c = 300, N = 12\}
, DistProp );\nevalf(%);\ntesteq( Left = Right);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 89 "subs( \{a[k] = (2*k +1)/k^3, c = -7, N = 30\}
, DistProp );\nevalf(%);\ntesteq( Left = Right);" }}}{PARA 0 "" 0 "" 
{TEXT -1 3 "\n\n\n" }{TEXT 273 17 "................ " }{TEXT 272 24 "T
he Associative Property" }{TEXT 274 98 " .............................
...................................................................." 
}{TEXT -1 76 "\n\n\nHere is the associate property of addition, and so
me examples of its use." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "
AssocProp :=  Sum( a[k] + b[k]  , k = 1..N) \n              = Sum( a[k
]  , k = 1..N) + Sum( b[k]  , k = 1..N);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "subs( \{a[k]=5*k-7, b[k]=3-11*k, N = 125\}, AssocProp
); \nevalf(%);\ntesteq( Left = Right);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "subs( \{a[k] = 8*k + 21, b[k]=13*k - 9, N = 56\}, Ass
ocProp );\nevalf(%);\ntesteq( Left = Right);" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 1 " " }{TEXT 265 18 "4. Series Formulas" }}{PARA 0 "" 0 
"" {TEXT -1 52 "\nWhen adding a constant, there is a simple formula :
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 35 "Sum( 1, k = 1..n): % = value(%);   " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 33 "Sum( C, k = 1..n): % = value(%); " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum( 17, k = 1..2000): % = value(%)
;" }}}{PARA 0 "" 0 "" {TEXT -1 386 "\n\nThere are formulas to compute \+
the sum of consecutive integers, squares, cubes, etc. You can find man
y of these formulae in your textbook or a reference book, or let Maple
 find the formula for you. To get an attractive formula, we will compu
te the sum, simplify the result, and factor that result.\n\nYou might \+
know this formula already. This is the sum of integers : 1 + 2 + 3 + .
.. + n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "`Sum of Integers`;
 \nSum( k, k = 1..n): % = value(%);  \nFormula := factor(simplify(%));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs( \{ n = 100\}, For
mula );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs( \{ n = 2^1
0\}, Formula );" }}}{PARA 0 "" 0 "" {TEXT -1 61 "\n\nHere is a formula
 for the squares of the natural numbers. \n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 89 "`Sum of Squares`;  \nSum( k^2, k = 1..n): % = value
(%);   \nFormula := factor(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "subs( \{ n = 100\}, Formula );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "subs( \{ n = 2^10\}, Formula );" }}}{PARA 0 "
" 0 "" {TEXT -1 77 "\nHere are sums of higher powers. Each can be simp
lified into a simpler form.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 76 "`Sum of Cubes`;  \nSum( k^2, k = 1..n): % = value(%);   \nfactor
(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "`Sum of 4
th Powers`;  \nSum( k^2, k = 1..n): % = value(%);   \nfactor(simplify(
%));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 
"Maple is really wonderful at computing the formulas. Here is one for \+
adding eighth powers of integers." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "`Sum of 4th Powers`;  \nSum( k^8, k = 1..n): % = valu
e(%);   \nfactor(simplify(%));" }}}{PARA 0 "" 0 "" {TEXT -1 173 "\n\n
\nHere are some other interesting sums... the squares of odds, cubes o
f negative odds, 5th powers of negative even numbers, etc. There are f
ormulas for each of these cases!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 99 "`Sum of Squares of Odd Numbers`;  \nSum( (2*k-1)^2, k = 1..n):
 % = value(%);   \nfactor(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 109 "`Sum of Cubes of Negative Odd Numbers`;  \nSum( (-(2
*k-1))^3, k = 1..n): % = value(%);   \nfactor(simplify(%));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "`Sum of 5th Powers of Negat
ive Even Numbers`;  \nSum( (-2*k)^5, k = 1..n): % = value(%);   \nfact
or(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "`Sum o
f Squares of Odd Numbers and Cubes of Evens`;  \nSum( (2*k-1)^2 + (2*k
)^3, k = 1..n): % = value(%);   \nfactor(simplify(%));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "`Sum of Cubes of an Arithmetic Seq
uence`;  \nSum( (7*k + 2)^3, k = 1..n): % = value(%);   \nfactor(simpl
ify(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "`Sum of Cubes \+
of a General Arithmetic Sequence`;  \nSum( (a*k + b)^3, k = 1..n): % =
 value(%);   \nfactor(simplify(%));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 1 " " }{TEXT 266 45 "5. Infinite Series - Convergence & Divergence
" }}{PARA 0 "" 0 "" {TEXT -1 97 "\nWhen you add up an infinite number \+
of numbers, you are 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..infinity);   % = value(%);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "However, the result is
 not necessarily infinite. If the numbers get small quickly enough, th
e 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..infinit
y);   % = value(%);" }}}{PARA 0 "" 0 "" {TEXT -1 75 "\nAnother strange
 thing that can occur is a sum that oscillates infinitely.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "Sum( (-1)^k, k = 1..4):   %
 = value(%);\nSum( (-1)^k, k = 1..5):   % = value(%);\nSum( (-1)^k, k \+
= 1..400):   % = value(%);\nSum( (-1)^k, k = 1..401):   % = value(%);
" }}}{PARA 0 "" 0 "" {TEXT -1 76 "\nWhen a sum adds up to infinity, ne
gative infinity or oscillates, we say it " }{TEXT 268 8 "diverges" }
{TEXT -1 64 ". If it gets closer and closer to a particular number we \+
say it " }{TEXT 267 9 "converges" }{TEXT -1 58 ". Surprisingly, there \+
are many convergent infinite series." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "Sum( 1/k^2, k = 1..infinity):   % = evalf(value(%));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Sum( 1/k!, k = 0..infin
ity):   % = evalf(value(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "Sum( 1/exp(k), k = 1..infinity):   % = evalf(value(%));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Sum( ((-2)^j)/j!, j = 0..inf
inity):   % = evalf(value(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 56 "Sum( sin(j)/j!, j = 0..infinity):   % = evalf(value(%));" }}}
{PARA 0 "" 0 "" {TEXT -1 121 "\nSometimes it's hard to tell at first. \+
Even if we add 10,000 member of this series, we still have a number le
ss than 10!\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Sum( 1/k, k
 = 1..100);   % = evalf(value(%));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "Sum( 1/k, k = 1..1000);   % = evalf(value(%));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum( 1/k, k = 1..10000);   %
 = evalf(value(%));" }}}{PARA 0 "" 0 "" {TEXT -1 77 "\n\nHowever, look
s can be decieving. This series actually diverges to infinity." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum( 1/k, k = 1..infinity); \+
  % = evalf(value(%));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 275 19 "6. Graphs of Series" }
}{PARA 0 "" 0 "" {TEXT -1 209 "\nWe can also visualize a series. The o
range line shows the sum, while the blue line shows the value of seque
nce. In this case, being an arithmetic sequence, the sum is getting la
rger and larger... to infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a := k -> 3*k - 2;   " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "plot( [a(floor(x)), sum( a(
j), j = 1..floor(x))] , \n      x = 1..10, thickness=2, linestyle = [3
,1], color = [blue,coral] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 124 "plot( [a(floor(x)), sum( a(j), j = 1..floor(x))] , \n      x = \+
1..30, thickness=2, linestyle = [3,1], color = [blue,coral] );" }}}
{PARA 0 "" 0 "" {TEXT -1 198 "\nNow let's look at a geometric series .
.. which converges. We can see that the series gets larger and larger,
 but it seems to be leveling out - because the sequence is getting ver
y small, very fast." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a := \+
k -> (2/3)^k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "plot( [a(
floor(x)), sum( a(j), j = 1..floor(x))] , \n      x = 1..10, thickness
=2, linestyle = [3,1], color = [blue,coral] );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 124 "plot( [a(floor(x)), sum( a(j), j = 1..floor(x
))] , \n      x = 1..30, thickness=2, linestyle = [3,1], color = [blue
,coral] );" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nHere are some other conv
ergent sequences." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a := k \+
-> 1/k^2;" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a := k
 -> 10*(7/8)^k;" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "a := k -> ((-1)^(k+1))/k^(9/8);" }}}{PARA 0 "" 0 "" {TEXT -1 
35 "\nHere are some divergent sequences." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "a := k -> (5 + (-k)*(-1) );            SumPlot(a);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a := k -> (8/7)^k ;       \+
             SumPlot(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"a := k -> (-1)^k ;            SumPlot(a);" }}}{PARA 0 "" 0 "" {TEXT 
-1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 14 "7.
 Double Sums" }}{PARA 0 "" 0 "" {TEXT -1 85 "\nA more complicated situ
ation is the double sum - where there is a sum within a sum. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
80 "S  := Sum( Sum(i^2, i = 1..N), j = 1..M);   \nSF := factor( simpli
fy( value(%)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs( \{
N = 2, M = 3 \}, S = SF);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "subs( \{N = 5, M = 10\}, S = SF);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "subs( \{N = 50, M = 100\}, S = SF);" }}}{PARA 0 "" 0 
"" {TEXT -1 71 "\nNotice that if we interchanged the two sums, we get \+
different results." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "S  := \+
Sum( Sum(j^2, j = 1..M), i = 1..N);   \nSF := factor( simplify( value(
%)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs( \{N = 2, M =
 3 \}, S = SF);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs( \{
N = 5, M = 10\}, S = SF);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "It can also get \+
more complicated if the limit of the inner sum depends on the index of
 the outer sum. You should try these by hand.\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 80 "S  := Sum( Sum(k^2, k = 1..m), m = 1..M);   \n
SF := factor( simplify( value(%)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "subs( \{N = 2, M = 3\}, S = SF);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "subs( \{N = 5, M = 10\}, S = SF);" }}}
{EXCHG }}{PARA 0 "" 0 "" {TEXT 259 36 "\n         \251 2002 Waterloo M
aple Inc " }}}{MARK "0 1" 40 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }