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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 49 "\n\011\nPointwise Converg
ence\n\011\nPointwise Convergence." }}{PARA 0 "" 0 "" {TEXT -1 13 "A s
equence \{ " }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }
{TEXT -1 45 "\} of functions from a set S to R  is said to " }{TEXT 
256 18 "converge pointwise" }{TEXT -1 52 " to a function  f  if for ev
ery  x  in S  we have  \{" }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6
#%\"xG" }{TEXT -1 48 "\} converges to  f(x) as  n  tends to infinity.
\011\n" }}{PARA 0 "" 0 "" {TEXT 258 19 "Bounded Convergence" }{TEXT 
-1 15 "\n\011\nA sequence \{" }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"
nG6#%\"xG" }{TEXT -1 49 "\}  of functions from a set  S  to  R  is sai
d to " }{TEXT 259 18 "converge boundedly" }{TEXT -1 22 " to a function
 f  if \{" }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT 
-1 82 "\}  converges pointwise to  f  and it is possible to find a num
ber  K  such that  |" }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"x
G" }{TEXT -1 39 "| <= K for every x in S and every  n.\011\n" }}{PARA 
0 "" 0 "" {TEXT 260 19 "Uniform Convergence" }{TEXT -1 15 "\n\011\nA s
equence \{" }{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT 
-1 48 "\} of functions from a set  S  to  R  is said to " }{TEXT 261 
18 "converge uniformly" }{TEXT -1 28 " to a function  f  if  sup |" }
{XPPEDIT 18 0 "f[n](x)" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 3 " - " }
{TEXT 265 4 "f(x)" }{TEXT -1 42 " | tends to  0  as  n tends to infini
ty.\011\n" }}{PARA 0 "" 0 "" {TEXT 262 10 "Exercise 1" }{TEXT -1 46 ":
 Demonstrate that the sequence of functions  " }{XPPEDIT 18 0 "f(x,n)=
n*x*exp(-n*x)" "6#/-%\"fG6$%\"xG%\"nG*(F(\"\"\"F'F*-%$expG6#,$*&F(F*F'
F*!\"\"F*" }{TEXT -1 60 "  is pointwise convergent to 0 but not unifor
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6\"F)F)*(9%\"\"\"9$F,-%$expG6#,$*&F+F,F-F,!\"\"F,F)F)6\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(x,n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*(%\"nG\"\"\"%\"xGF%-%$expG6#,$*&F$F%F&F%!\"\"F%" }}}
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 10 "Exercise 2" }
{TEXT -1 59 ": Demonstrate by animation that the sequence of functions
  " }{XPPEDIT 18 0 "f(x,n)=n*x*exp(-n^2*x)" "6#/-%\"fG6$%\"xG%\"nG*(F(
\"\"\"F'F*-%$expG6#,$*&F(\"\"#F'F*!\"\"F*" }{TEXT -1 26 " converges un
iformly to 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "f:=proc(x,k) k*x*exp((-k^2)*x) end;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"kG6\"F)F)*(9%\"\"\"9$F,-%$expG
6#,$*&F+\"\"#F-F,!\"\"F,F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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{PARA 0 "" 0 "" {TEXT 263 10 "Exercise 3" }{TEXT -1 59 ": Demonstrate \+
by animation that the sequence of functions  " }{XPPEDIT 18 0 "f(x,n)=
n^2*x*exp(-n*x)" "6#/-%\"fG6$%\"xG%\"nG*(F(\"\"#F'\"\"\"-%$expG6#,$*&F
(F+F'F+!\"\"F+" }{TEXT -1 39 " converges pointwise but not boundedly.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "f:=proc(x,k) k^2*x*exp(-k*x) end;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"kG6\"F)F)*(9%\"\"#9$\"\"\"-%$expG6#,
$*&F+F.F-F.!\"\"F.F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for k f
rom 1 by 1 to 30 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f.k:= plot \+
(\{f(x,30),f(x,k)\}, x=0..1, y=0..15,thickness=2):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "pl
ots[display]([seq(f.k, k=1..30)], insequence=true);" }}{PARA 13 "" 1 "
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