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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Infinite Series" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 12 "Introduction" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Vari
ous Infinite Series" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Infinite s
eries are treated by Maple just as we write them in class, i.e., as su
ms where the upper limit is infinity." }}{PARA 0 "" 0 "" {TEXT -1 75 "
Maple can evaluate some sums, and can't do others (just as with integr
als)." }}{PARA 0 "" 0 "" {TEXT -1 14 "It can handle " }{TEXT 257 16 "g
eometric series" }{TEXT -1 22 " without any problems:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 58 "Sum((1/2)^n, n=0..infinity) = sum((1/2)^n, n=0..
infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "For the " }{TEXT 
261 25 "divergent Harmonic series" }{TEXT -1 55 " Maple shows us that \+
this series diverges by returning " }{XPPEDIT 18 0 "infinity" "6#%)inf
inityG" }{TEXT -1 14 " as an answer:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 50 "Sum(1/n, n=1..infinity) = sum(1/n, n=1..infinity);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 32 "Here is another infinite series." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(1/(4*n^2-1), n=1..infinity) = s
um(1/(4*n^2-1), n=1..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
34 "In class we would treat this as a " }{TEXT 262 18 "telescoping ser
ies" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "We can do telescop
ing series in Maple, also." }}{PARA 0 "" 0 "" {TEXT -1 43 "Let's consi
der the previous example, i.e., " }{XPPEDIT 18 0 "sum(1/(4*n^2-1), n=1
..infinity)" "6#-%$sumG6$*&\"\"\"F',&*&\"\"%F'*$%\"nG\"\"#F'F'F'!\"\"F
./F,;F'%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "Let
's first compute a " }{HYPERLNK 17 "partial fractions decomposition" 
2 "convert,parfrac" "" }{TEXT -1 35 " of the general term of the serie
s:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "convert(1/(4*n^2-1), parfrac,
 n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Next, we turn this into a
 function. This makes the following work easier." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "a := unapply(%, n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 31 "Now we are ready to define the " }{XPPEDIT 18 0 "n" "6#%\"nG" }
{TEXT -1 36 "-th partial sum (also as a function)" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "s := n -> sum(a(k), k=1..n);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 26 "and compute a few of them." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "s[1] = s(1);\ns[2] = s(2);\ns[3] = s(3);\ns[4] = s(4)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Unfortunately Maple simplif
ies the expressions right away, so we do not see the \"telescoping eff
ect\"." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "n" "6#%\"n
G" }{TEXT -1 18 "-th partial sum is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "s[n] = s(n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "and the limit
 as " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 65 " tends to infinity (a
nd therefore the sum of the infinite series)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "'s' = limit(s(n), n=infinity);" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 6 "A Game" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "We en
d with a game:\n" }{TEXT 258 7 "Problem" }{TEXT -1 114 ": Suppose Jack
 and Jill are rolling a fair six-sided die in turn until one of them w
ins by getting the first six. " }}{PARA 0 "" 0 "" {TEXT -1 75 "If Jill
 rolls first, calculate the probability that she will win the game.\n
" }{TEXT 259 8 "Solution" }{TEXT -1 85 ": Because the die is fair, the
 probability that Jill gets a six on the first roll is " }{XPPEDIT 18 
0 "1/6" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 73 "The probability that she gets the game's first six on the
 second roll is " }{XPPEDIT 18 0 "(5/6)^2*(1/6)" "6#*&*&\"\"&\"\"\"\"
\"'!\"\"\"\"#*&F&F&F'F(F&" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 
-1 60 "This number is determined as the product of the probability " }
{XPPEDIT 18 0 "(5/6)^2" "6#*$*&\"\"&\"\"\"\"\"'!\"\"\"\"#" }{TEXT -1 
79 " that neither Jill nor Jack rolls a six in the first round and the
 probability " }{XPPEDIT 18 0 "1/6" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT 
-1 43 " that Jill rolls a six in the second round." }}{PARA 0 "" 0 "" 
{TEXT -1 19 "Jill's probability " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT 
-1 45 " of getting the first six in the game is the " }{TEXT 260 3 "su
m" }{TEXT -1 108 " of her probabilities of getting it in the first rou
nd, in the second round, in the third round, and so on. " }}{PARA 0 "
" 0 "" {TEXT -1 3 "So " }}{PARA 0 "" 0 "" {TEXT -1 18 "               \+
   " }{XPPEDIT 18 0 "p = 1/6 + (5/6)^2*(1/6) + (5/6)^2*(5/6)^2*(1/6)" 
"6#/%\"pG,(*&\"\"\"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)F'F'" }{TEXT -1 30 " + ...\n         \+
            = " }{XPPEDIT 18 0 "1/6" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT 
-1 2 " [" }{XPPEDIT 18 0 "1 + (5/6)^2 + (5/6)^4" "6#,(\"\"\"F$*$*&\"\"
&F$\"\"'!\"\"\"\"#F$*$*&F'F$F(F)\"\"%F$" }{TEXT -1 8 " + ... ]" }}
{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "
p := (1/6)*sum(((5/6)^2)^n, n=0..infinity);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 121 "We see that Jill - by going first - has a better chanc
e of winning the game than Jack (whose probability, of course,  is " }
{XPPEDIT 18 0 "5/11" "6#*&\"\"&\"\"\"\"#6!\"\"" }{TEXT -1 2 ")." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 13 "Assignment 10" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.1
:" }}{PARA 0 "" 0 "" {TEXT -1 32 "Use Maple to evaluate the series" }}
{PARA 0 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "sum((exp(1)/Pi)^n, n=
1..infinity)" "6#-%$sumG6$)*&-%$expG6#\"\"\"F+%#PiG!\"\"%\"nG/F.;F+%)i
nfinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "b) " }
{XPPEDIT 18 0 "sum((6*n^2+2*n-1)/(n*(n+1)*(4*n^2-1)), n=1..infinity)" 
"6#-%$sumG6$*&,(*&\"\"'\"\"\"*$%\"nG\"\"#F*F**&F-F*F,F*F*F*!\"\"F**(F,
F*,&F,F*F*F*F*,&*&\"\"%F**$F,F-F*F*F*F/F*F//F,;F*%)infinityG" }{TEXT 
-1 1 "," }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.2:" }}{PARA 0 "" 0 
"" {TEXT -1 89 "For the series in 1b) repeat the  partial fractions ap
proach as used in the introduction." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 5 "Ex.3:" }}{PARA 0 "" 0 "" {TEXT -1 94 "Determine experimentally (
by trial and error) how many terms of the divergent Harmonic series " 
}{XPPEDIT 18 0 "sum(1/n, n=1..infinity)" "6#-%$sumG6$*&\"\"\"F'%\"nG!
\"\"/F(;F'%)infinityG" }{TEXT -1 55 " are needed so that the partial s
um exceeds 5, 10, 20? " }}{PARA 0 "" 0 "" {TEXT -1 8 "Comment." }}
{PARA 0 "" 0 "" {TEXT -1 10 "Hint: Use " }{HYPERLNK 17 "evalf" 2 "eval
f" "" }{TEXT -1 34 " when evaluating the partial sums." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 5 "Ex.4:" }}{PARA 0 "" 0 "" {TEXT -1 145 "Sup
pose that the government spends $1 billion and each recipient of a fra
ction of this wealth spends 90% of the dollars that he or she receives
. " }}{PARA 0 "" 0 "" {TEXT -1 84 "In turn, the secondary recipients s
pend 90% of the dollars they receive, and so on. " }}{PARA 0 "" 0 "" 
{TEXT -1 99 "How much total spending thereby results from the original
 injection of $1 billion into the economy?" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 5 "Ex.5:" }}{PARA 0 "" 0 "" {TEXT -1 124 "It is said that t
he ancient Etruscans played dice using a dodecahedral die having 12 pe
ntagonal faces numbered 1 through 12." }}{PARA 0 "" 0 "" {TEXT -1 32 "
Here's a plot of a dodecahedron:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 66 "with(plottools):plots[display](dodecahedron([0,0,0]),style=pat
ch);" }}}{PARA 0 "" 0 "" {TEXT -1 102 "One could simulate such a die b
y drawing a random card from a deck of 12 cards numbered 1 through 12.
 " }}{PARA 0 "" 0 "" {TEXT -1 43 "For this problem, let's think of a d
eck of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 26 " cards numbered 1 \+
through " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 31 "For your own personal value of " }{XPPEDIT 18 0 "k" 
"6#%\"kG" }{TEXT -1 75 ", begin with the largest digit in the sum of t
he digits of your birthdate. " }}{PARA 0 "" 0 "" {TEXT -1 22 "This is \+
your value of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 90 " unless thi
s digit is less than 5, in which case subtract it from 10 to get your \+
value of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 52 "(Do this in Maple so that I can see your birthdate.)
" }}{PARA 0 "" 0 "" {TEXT -1 58 "a) John and Mary draw alternately fro
m a shuffled deck of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 8 " card
s. " }}{PARA 0 "" 0 "" {TEXT -1 59 "The first one to draw an ace - the
 card numbered 1 - wins. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Assume that \+
John draws first. " }}{PARA 0 "" 0 "" {TEXT -1 95 "Use the geometric s
eries approach illustrated in the introduction to calculate the probab
ility " }{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 47 " that John wins, an
d similarly the probability " }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 
17 " that Mary wins. " }}{PARA 0 "" 0 "" {TEXT -1 11 "Check that " }
{XPPEDIT 18 0 "J+M = 1" "6#/,&%\"JG\"\"\"%\"MGF&F&" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 62 "b) Now John, Mary, and Paul draw alternat
ely from the deck of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 8 " card
s. " }}{PARA 0 "" 0 "" {TEXT -1 115 "Calculate separately their respec
tive probabilities of winning, given that John draws first and Mary dr
aws second. " }}{PARA 0 "" 0 "" {TEXT -1 11 "Check that " }{XPPEDIT 
18 0 "J+M+P=1" "6#/,(%\"JG\"\"\"%\"MGF&%\"PGF&F&" }{TEXT -1 1 "." }}}}
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