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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 270 22 "Section 8.4  Example 6" }
{TEXT 272 4 ":   " }{TEXT -1 1 " " }{TEXT 271 57 "Illustration of the \+
Alternating Series Estimation Theorem" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 20 "Define the sequence " }{TEXT 256 1 "a
" }{TEXT -1 1 "(" }{TEXT 257 1 "n" }{TEXT -1 12 ") to be the " }{TEXT 
258 1 "n" }{TEXT -1 28 "th term of the given series." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "a := n -> (-1)^n/(2*n)!;" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"nG6\"6$%)operatorG%&arrowG
F(*&)!\"\"9$\"\"\"-%*factorialG6#,$F/\"\"#F.F(F(F(" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "We need to write
 out a few terms of the sequence and determine the first one which has
 4 zeroes immediately following the decimal point " }{TEXT 264 6 "(Why
?)" }{TEXT -1 23 ".  Indeed, observe that" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(a(n),n=1..4); n:='n';" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6&#!\"\"\"\"##\"\"\"\"#C#F$\"$?(#F'\"&?
.%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" }}}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Here, we simply evaluate \+
the above fractions with floating point accuracy." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(a(0));
" }}{PARA 11 "" 1 "" {TEXT -1 21 "No Way !!!!!!        " }{XPPMATH 20 
"6#$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(a(
1));" }}{PARA 11 "" 1 "" {TEXT -1 15 "Nope!!!        " }{XPPMATH 20 "6
#$!+++++]!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(a(2))
;" }}{PARA 11 "" 1 "" {TEXT -1 32 "Still not good enough.          " }
{XPPMATH 20 "6#$\"+nmmmT!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "evalf(a(3));" }}{PARA 11 "" 1 "" {TEXT -1 39 "Still not, but getti
ng better.         " }{XPPMATH 20 "6#$!+*))))))Q\"!#7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(a(4));" }}{PARA 11 "" 1 "" 
{TEXT -1 25 "Good enough!!!           " }{XPPMATH 20 "6#$\"+I(e,[#!#9
" }}}{PARA 256 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "No
w, we shall use " }{TEXT 259 1 "a" }{TEXT -1 168 "(4) in the Alternati
ng Series Estimation Theorem.  The claim is that S4 (sum of the first \+
four terms) should be within 4 decimal places of the actual sum of the
 series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "Sum(a(n), n=0..4); S4 := evalf(%);" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"
\"%\"nG\"\"\"-%*factorialG6#,$F)\"\"#F(/F);\"\"!\"\"%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#S4G$\"+%zDIS&!#5" }}}{PARA 11 "" 1 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Consequently, if S" }{TEXT 
260 1 " " }{TEXT -1 62 "denotes the sum of the original series, we can
 conclude that  " }}{PARA 0 "" 0 "" {TEXT -1 71 "                     \+
                 0.5403025794 - 0.000025   <=    S" }{TEXT 261 6 "    \+
<=" }{TEXT -1 28 "    0.5403025794 + 0.000025." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 262 6 "Remark" }{TEXT -1 32 ":  Suppose that we want to find \+
" }{TEXT 263 1 "S" }{TEXT -1 217 " accurate to 20 decimal places.  We \+
follow the same scheme, but we need not start at the first term, since
 it is unlikely that the very first term of the series has 20 zeroes i
mmediately following the decimal point.  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "seq(a(n),n=8..14); n='n';" }{TEXT -1 74 "            \+
 ******* Not exactly helpful.  Just do what follows!! ********" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)#\"\"
\"\"/+!)))*yA4##!\"\"\"1+!GdqtBS'#F$\"4++kw\"3?!HV##F'\"7++o2wxF2+C6#F
$\"9++OR%RKt,%[/i#F'\"<+++%eNcgE6Y\"H.%#F$\"?+++/:]gQr6YM))[I" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"nGF$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 215 "Note that it makes sense
 to start at the term with the highest index in our list since if it d
oes not have the desired degree of accuracy, then no term in our list \+
that comes before it can have the desired accuracy " }{TEXT 265 6 "(Wh
y?)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "evalf(a(14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+P#*))zK!#R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(a(
12));" }}{PARA 11 "" 1 "" {TEXT -1 37 "Still more accuracy than desire
d.    " }{XPPMATH 20 "6#$\"+rvt6;!#L" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "evalf(a(11));" }}{PARA 11 "" 1 "" {TEXT -1 40 "Probab
ly the best we can hope for.      " }{XPPMATH 20 "6#$!+#R\"z'*))!#J" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(a(10));" }}{PARA 11 
"" 1 "" {TEXT -1 26 "YEPPPPP !!!!!!!!!!!!!   \010\010" }{XPPMATH 20 "6
#$\"+BwJ5T!#G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 74 "Now, draw the conclusions as in the first set of calcu
lations to estimate " }{TEXT 266 0 "" }{TEXT -1 0 "" }{TEXT 267 1 "S" 
}{TEXT -1 14 " in this case." }}}{PARA 257 "" 1 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 268 7 "Problem" }{TEXT -1 37 ":   Do the same fo
r the series whose " }{TEXT 269 1 "n" }{TEXT -1 25 "th term is the fol
lowing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "b(n) := n -> (-3)^(n-1)/sqrt(n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"bG6#%\"nGRF&6\"6$%)operatorG%&arrowGF)*&)!\"$,&9$
\"\"\"!\"\"F2F2-%%sqrtG6#F1F3F)F)F)" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{MARK "28" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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