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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 16 "Maple Project 10" }}
{PARA 0 "" 0 "" {TEXT -1 63 "Here's how we do absolute values on Maple
: to find | -6 |, type" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ab
s(-6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Now let's review how to
 create functions and take limits." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "a:=n->1/n^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "limit(a(n),n=infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 
"and plot 2 sequences on the same plot (one denoted by a circle, the o
ther by a cross, so we can tell them apart)." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "A:=plot( [ seq([n,a(n)],n=1..20) ] , style=point):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "B:=plot( [ seq([n,a(n)^2],
n=1..20) ] , style=point, symbol=cross):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "display([A,B],title=\"Man.  It's just sweet.\");" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "1) Determine convergence or diverg
ence of the series  " }{XPPEDIT 18 0 "sum(n/(n^2-1),n = 2 .. infinity)
;" "6#-%$sumG6$*&%\"nG\"\"\",&*$F'\"\"#F(\"\"\"!\"\"F-/F';\"\"#%)infin
ityG" }{TEXT -1 66 "  using the following tests.  State your conclusio
n for each test." }}{PARA 0 "" 0 "" {TEXT -1 81 " a) Divergence Test  \+
     b) Root Test       c) Ratio Test       d) Integral Test" }}{PARA 
0 "" 0 "" {TEXT -1 42 "2) Determine convergence or divergence of " }
{XPPEDIT 18 0 "sum(product(2*k-1,k = 1 .. n)/(e^n*n!),n = 1 .. infinit
y);" "6#-%$sumG6$*&-%(productG6$,&*&\"\"#\"\"\"%\"kGF-F-\"\"\"!\"\"/F.
;\"\"\"%\"nGF-*&)%\"eGF4F--%*factorialG6#F4F-F0/F4;\"\"\"%)infinityG" 
}{TEXT -1 58 "    by using an appropriate test on Maple.  The notation
  " }{XPPEDIT 18 0 "product(2*k-1,k = 1 .. n)" "6#-%(productG6$,&*&\"
\"#\"\"\"%\"kGF)F)\"\"\"!\"\"/F*;\"\"\"%\"nG" }{TEXT -1 158 " means 1*
3*5*7*...*(2n-3)*(2n-1).     (Hint: to multiply things together, one c
an use the \"product\" command in exactly the same way we use the \"su
m\" command.)" }}{PARA 0 "" 0 "" {TEXT -1 54 "3) Determine convergence
 or divergence of the series  " }{XPPEDIT 18 0 "sum((n^3+n)/(n^6-n^2+1
),n = 1 .. infinity);" "6#-%$sumG6$*&,&*$%\"nG\"\"$\"\"\"F)F+F+,(*$F)
\"\"'F+*$F)\"\"#!\"\"\"\"\"F+F1/F);\"\"\"%)infinityG" }{TEXT -1 68 "  \+
 using the Limit Comparison Test.  (Hint: for large n, what does  " }
{XPPEDIT 18 0 "(n^3+n)/(n^6-n^2+1);" "6#*&,&*$%\"nG\"\"$\"\"\"F&F(F(,(
*$F&\"\"'F(*$F&\"\"#!\"\"\"\"\"F(F." }{TEXT -1 13 "  look like?)" }}
{PARA 0 "" 0 "" {TEXT -1 23 "4) Plot the sequences  " }{XPPEDIT 18 0 "
a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sin(n)^2/(n
^(3/2));" "6#*&*$)-%$sinG6#%\"nG\"\"#\"\"\"F+)%\"nG*&\"\"$\"\"\"\"\"#!
\"\"!\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"
nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(n^(3/2));" "6#*&\"\"\"\"\"\")
%\"nG*&\"\"$F%\"\"#!\"\"F+" }{TEXT -1 194 "  from n=20 to 150 on the s
ame plot.  Assuming what we see continues off to infinity (easily chec
ked by hand), can we use the Comparison Test to say anything about the
 convergence of the series " }{XPPEDIT 18 0 "sum(a[n],n = 1 .. infinit
y);" "6#-%$sumG6$&%\"aG6#%\"nG/%\"nG;\"\"\"%)infinityG" }{TEXT -1 28 "
 ?  Now plot the sequences  " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" 
}{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(n+sqrt(n));" "6#*&\"\"\"\"\"\",&%
\"nG\"\"\"-%%sqrtG6#F'F(!\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "b
[n];" "6#&%\"bG6#%\"nG" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "1/n;" "6#*&
\"\"\"\"\"\"%\"nG!\"\"" }{TEXT -1 184 "  again from n=20 to 150 on the
 same plot.  Assuming what we see continues (easily checked by hand), \+
can we use the Comparison Test to say anything about the convergence o
f the series " }{XPPEDIT 18 0 "sum(a[n],n = 1 .. infinity);" "6#-%$sum
G6$&%\"aG6#%\"nG/%\"nG;\"\"\"%)infinityG" }{TEXT -1 3 " ? " }}}}{MARK 
"10 1 0" 4 }{VIEWOPTS 1 1 0 1 1 1803 }