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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Convergence of sequences of funct
ions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Ca
nada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version:  26.3.2007 " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar
t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 
"" 0 "" {TEXT -1 60 "Pointwise and uniform convergence of a sequence o
f functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 
"" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" 
}{TEXT -1 3 " , " }{XPPEDIT 18 0 "n=1,2,3,` . . . `" "6&/%\"nG\"\"\"\"
\"#\"\"$%(~.~.~.~G" }{TEXT -1 84 ", be a sequence of functions defined
 on an interval I in the set of real numbers |R." }}{PARA 0 "" 0 "" 
{TEXT -1 72 "For our purposes, the interval will usually be either a c
losed interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 35 
" or the whole set of real numbers. " }}{PARA 0 "" 0 "" {TEXT -1 66 "W
e consider some different notions of convergence of the sequence " }
{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 15 " to a functio
n " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 33 ", also defined on t
he interval I." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Pointw
ise convergence of a sequence of functions" }}{PARA 0 "" 0 "" {TEXT 
-1 26 "The sequence of functions " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG
6#%\"nG" }{TEXT -1 1 " " }{TEXT 261 19 "converges pointwise" }{TEXT 
-1 4 " to " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 31 " on I if and
 only if, for each " }{TEXT 264 1 "x" }{TEXT -1 37 " in I, the sequenc
e of real numbers  " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%
\"xG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "phi(x);" "6#-%$phi
G6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "In detail: fo
r each positive number " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }
{TEXT -1 22 ", there is an integer " }{TEXT 267 1 "N" }{TEXT -1 38 ", \+
(which, in general, depends on both " }{TEXT 265 1 "x" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 12 ") such that
 " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "abs(phi[n](x)-p
hi(x)) < epsilon;" "6#2-%$absG6#,&-&%$phiG6#%\"nG6#%\"xG\"\"\"-F*6#F.!
\"\"%(epsilonG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 " for all
 " }{TEXT 266 1 "n" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "n>N" "6#2%\"N
G%\"nG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 
24 "________________________" }{TEXT 260 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Uniform converge
nce of a sequence of functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The sequence of functions " }
{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 1 " " }{TEXT 261 
19 "converges uniformly" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "phi" "6#%$
phiG" }{TEXT -1 60 " on the interval I if and only if, for each positi
ve number " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 22 ", th
ere is an integer " }{TEXT 268 1 "N" }{TEXT -1 19 ", which depends on \+
" }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 14 ", (but not on \+
" }{TEXT 269 1 "x" }{TEXT -1 12 ") such that " }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "abs(phi[n](x)-phi(x)) < epsilon;" "6#2
-%$absG6#,&-&%$phiG6#%\"nG6#%\"xG\"\"\"-F*6#F.!\"\"%(epsilonG" }}
{PARA 0 "" 0 "" {TEXT -1 9 " for all " }{TEXT 271 1 "n" }{TEXT -1 6 " \+
with " }{XPPEDIT 18 0 "n>N" "6#2%\"NG%\"nG" }{TEXT -1 9 " and all " }
{TEXT 270 1 "x" }{TEXT -1 7 " in I. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
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Fbt$\"3%)z*eNMNU<)F-7$$\"3'4+]7y#=o'*F-$\"3@([NT44a.)F-7$Fgt$\"3-IQCH0
B^yF-7$$\"3A+v$4^n)p5F1$\"3F_E%=5I#RwF-7$F\\u$\"3O*)38MWU1uF-7$$\"39+v
o/#3o<\"F1$\"3q^%=\"G.sSrF-7$Fau$\"3#=!Q.[?@')oF-7$$\"3;L$3_NJOG\"F1$
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\\QO'F-7$F[v$\"3G5$=Zp\"4iiF-7$$\"3mm\"H#oZ1\"\\\"F1$\"34Hf[\"***=-iF-
7$F`v$\"3eq[]r6flhF-7$$\"3%o;Hd!fX$f\"F1$\"3PU4**Gu;ThF-7$Fev$\"3n>`-d
62?hF-7$$\"3_Le*[t\\sp\"F1$\"3;n%f]E5@4'F-7$Fjv$\"3')3=Sc0.[gF-7$$\"3U
L$3_0j,!=F1$\"3gq\"o0h,)zfF-7$F_w$\"3u*R#\\\"z\"p))eF-7$Fdw$\"35=@y[C$
zm&F-7$$\"3*G$e*[$zV4?F1$\"3?\"4t]'e_`bF-7$Fiw$\"3%Hg,CzM6Y&F-7$$\"3G$
3_]p'>*3#F1$\"3*fV$p'f*eHaF-7$$\"3ym\"zW7@^6#F1$\"3Paa))=a$*3aF-7$$\"3
w3F>RL3G@F1$\"3Hn<'=95KS&F-7$$\"3H]i!RbX59#F1$\"3+\")[)*zU!3S&F-7$$\"3
#=z>'ox+a@F1$\"3a%eIAg))=S&F-7$F^x$\"3WSD*R+=mS&F-7$$\"3#Q$3xc9[$>#F1$
\"3TQ;K!z,#GaF-7$$\"3)QL3-$H**>AF1$\"3%eh3\"**)GlY&F-7$$\"3#R$ek.W]YAF
1$\"36\"*y5<Z>AbF-7$Fcx$\"31d&Q7SMaf&F-7$$\"3h+v=n(*fDBF1$\"3K$*G,z)R=
z&F-7$Fhx$\"3ge`6q\"f90'F-7$F]y$\"3Uo(==;JuL'F-7$Fby$\"3%zWkYQ>%emF-7$
Fgy$\"3?%*HuDAoaqF-7$F\\z$\"3mqxe7$y]Y(F-7$Faz$\"35UE)>B\"HKyF-7$Ffz$
\"3i]0K=)y?>)F-7$F[[l$\"3Y=(*f(4cPc)F-7$F`[l$\"3WCor\"3BU#*)F-7$Fe[l$
\"3OAeV5DMk#*F-7$Fj[l$\"3C:$R%4yq3'*F-7$F_\\l$\"3sm&p5v>j***F-7$Fd\\l$
\"31LMY(zb;/\"F1-%'COLOURG6&F[]l$\"*++++\"!\")$F*F*FfaoF\\hl-%%TEXTG6&
7$$\"$Q$F)$\"#8F]]lQ\"+6\"-Faao6&F[]lF*F*F*-%%FONTG6$%*HELVETICAG\"#5-
Fhao6&7$$\"$P$F)$\"\"*F]]lQ\"-F`boFaboFcbo-Fhao6&7$$\"$:$F)F]boQ\"fF`b
oFabo-Fdbo6$%'SYMBOLG\"#7-Fhao6&7$$\"#OF]]l$\"$J\"F)Q\"eF`boFaboFfco-F
hao6&7$Fcco$\"$7\"F)FecoFaboFfco-Fhao6&7$FccoF]coFecoFaboFfco-Fhao6&7$
$\"$e$F)$\"#\"*F)FadoFaboFfco-Fhao6&7$$!$6#F)$\"$+\"F)Q\"NF`boF`aoFcbo
-Fhao6&7$$!$D#F)$\"$0\"F)FecoF`aoFfco-%*AXESTICKSG6$F*F*-%+AXESLABELSG
6$Q\"xF`boQ!F`bo-%&TITLEG6#%4uniform~convergenceG-%%VIEWG6$;F\\foF]do;
$F]]lF]]l$\"#:F]]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 261 7 "Uniform" }{TEXT -1 21 " convergence implies " }
{TEXT 261 9 "pointwise" }{TEXT -1 34 " convergence, but the converse i
s " }{TEXT 260 3 "not" }{TEXT -1 6 " true." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "
phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 90 " is a sequence of conoinuous
 functions which converges uniformly to a continuous function " }
{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 23 " on a closed ionterval " 
}{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 34 "Then the sequence of real numbers " }{XPPEDIT 18 0 "
Int(phi[n](x),x = a .. b);" "6#-%$IntG6$-&%$phiG6#%\"nG6#%\"xG/F,;%\"a
G%\"bG" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "Int(phi(x),x = a
 .. b);" "6#-%$IntG6$-%$phiG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 1 "." }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 24 "______________________
__" }{TEXT 260 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "E
xplanation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 5 "Let  " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" 
}{TEXT -1 81 " be a sequence of functions which converges uniformly to
 the continuous function " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 
24 " on the closed interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 52 "Suppose we are given a \+
(small) positive real number " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG
" }{TEXT -1 14 ". Then choose " }{TEXT 272 1 "N" }{TEXT -1 8 " so that
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(phi[n](x)-phi
(x)) < epsilon/(b-a);" "6#2-%$absG6#,&-&%$phiG6#%\"nG6#%\"xG\"\"\"-F*6
#F.!\"\"*&%(epsilonGF/,&%\"bGF/%\"aGF2F2" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 8 "for all " }{TEXT 273 1 "n" }{TEXT -1 6 " with " }
{XPPEDIT 18 0 "n>N" "6#2%\"NG%\"nG" }{TEXT -1 9 " and all " }{TEXT 
274 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "a <=x" "6#1%\"aG%\"xG" 
}{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"abs(Int(phi[n](x),x = a .. b)-Int(phi(x),x = a .. b));" "6#-%$absG6#,
&-%$IntG6$-&%$phiG6#%\"nG6#%\"xG/F0;%\"aG%\"bG\"\"\"-F(6$-F,6#F0/F0;F3
F4!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=``" "6#/%!GF$" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "abs(Int(phi[n](x)-phi(x),x = a .. b))" "6#-%$abs
G6#-%$IntG6$,&-&%$phiG6#%\"nG6#%\"xG\"\"\"-F,6#F0!\"\"/F0;%\"aG%\"bG" 
}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "``
 <= Int(abs(phi[n](x)-phi(x)),x = a .. b);" "6#1%!G-%$IntG6$-%$absG6#,
&-&%$phiG6#%\"nG6#%\"xG\"\"\"-F.6#F2!\"\"/F2;%\"aG%\"bG" }{TEXT -1 2 "
  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= Int(epsil
on/(b-a),x = a .. b);" "6#1%!G-%$IntG6$*&%(epsilonG\"\"\",&%\"bGF*%\"a
G!\"\"F./%\"xG;F-F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = epsilon;" "6#/%!G%(epsilonG" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 45 "It follows that the sequence of real num
bers " }{XPPEDIT 18 0 "Int(phi[n](x),x = a .. b);" "6#-%$IntG6$-&%$phi
G6#%\"nG6#%\"xG/F,;%\"aG%\"bG" }{TEXT -1 15 "  converges to " }
{XPPEDIT 18 0 "Int(phi(x),x = a .. b);" "6#-%$IntG6$-%$phiG6#%\"xG/F);
%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Examples of pointwise and unifor
m convergence" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the sequen
ce of functions " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"
\"$%(~.~.~.~G" }{TEXT -1 26 ", defined on the interval " }{XPPEDIT 18 
0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 6 " by:  " }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "phi[n](x) = (1-x^2)^(1/2+1/(2*n))
;" "6#/-&%$phiG6#%\"nG6#%\"xG),&\"\"\"F-*$F*\"\"#!\"\",&*&F-F-F/F0F-*&
F-F-*&F/F-F(F-F0F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 " Th
e sequence " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 "
, " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.
~.~.~G" }{TEXT -1 12 ", converges " }{TEXT 261 9 "pointwise" }{TEXT 
-1 17 " to the function " }{XPPEDIT 18 0 "g(x) = sqrt(1-x^2);" "6#/-%
\"gG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
177 "phi := (n,x) -> (1-x^2)^(1/2+1/(2*n));\ng := x -> sqrt(1-x^2);\np
lot([g(x),phi(1,x),phi(2,x),phi(4,x),phi(8,x)],x=-1..1,\n  color=[blac
k,red,blue,green,magenta],linestyle=[2,1$4]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$phiGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)),&\"\"
\"F/*$)9%\"\"#F/!\"\",&#F/F3F/*&F6F/*&F/F/9$F4F/F/F)F)F)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrt
G6#,&\"\"\"F0*$)9$\"\"#F0!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 
430 224 224 {PLOTDATA 2 "6)-%'CURVESG6%7_o7$$!\"\"\"\"!$F*F*7$$!3-n;Hd
Nvs**!#=$\"3aIk@d'**oP(!#>7$$!3/MLe9r]X**F/$\"3^xD,?$RD/\"F/7$$!3/,](=
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "For examp
le, taking " }{XPPEDIT 18 0 "x = 1/2" "6#/%\"xG*&\"\"\"F&\"\"#!\"\"" }
{TEXT -1 15 ", the sequence " }{XPPEDIT 18 0 "phi[n](1/2);" "6#-&%$phi
G6#%\"nG6#*&\"\"\"F*\"\"#!\"\"" }{TEXT -1 14 " converges to " }
{XPPEDIT 18 0 "g(1/2) = sqrt(3)/2;" "6#/-%\"gG6#*&\"\"\"F(\"\"#!\"\"*&
-%%sqrtG6#\"\"$F(F)F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "phi := (n,x) -> (1-x^2)
^(1/2+1/(2*n)):\ng := x -> sqrt(1-x^2):\np1 := plot([g(1/2),[seq([n,ph
i(n,1/2)],n=1..15)]],0..15,\n  symbol=circle,style=[line,point],linest
yle=3,color=[cyan,blue]):\np2 := plot(phi(x,1/2),x=1..15,color=grey,li
nestyle=2):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Now we investigate whether the \+
sequence  " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 11 "
 converges " }{TEXT 261 9 "uniformly" }{TEXT -1 5 " to  " }{XPPEDIT 
18 0 "psi(x) = sqrt(1-x^2);" "6#/-%$psiG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F
'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "We can plo
t the difference between a function " }{XPPEDIT 18 0 "phi[n](x) = (1-x
^2)^(1/2+1/(2*n))" "6#/-&%$phiG6#%\"nG6#%\"xG),&\"\"\"F-*$F*\"\"#!\"\"
,&*&F-F-F/F0F-*&F-F-*&F/F-F(F-F0F-" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "g(x) = sqrt(1-x^2);" "6#/-%\"gG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"#
!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "For example, the
 maximum difference between " }{XPPEDIT 18 0 "phi[6](x);" "6#-&%$phiG6
#\"\"'6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sqrt(1-x^2);" "6#-%
%sqrtG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 15 " is about 0.05." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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ot(g(x)-phi(6,x),x=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 374 153 153 
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8rU*plMuA*F\\w7$$\"3i******\\2goPF/$\"379vW#fe[<\"F27$$\"3UKL$eR<*fTF/
$\"3!)*e3f/p&G9F27$$\"3m******\\)Hxe%F/$\"3.w\"*[*=YFt\"F27$$\"3ckm;H!
o-*\\F/$\"3c]&p%GEnV?F27$$\"3y)***\\7k.6aF/$\"3QxeK)yuIR#F27$$\"3#emmm
T9C#eF/$\"3%zw<F*=8dFF27$$\"33****\\i!*3`iF/$\"3m`J'H4H&fJF27$$\"3%QLL
L$*zym'F/$\"3_mu^2*\\Tc$F27$$\"3wKLL3N1#4(F/$\"31ChjMf$**)RF27$$\"3Nmm
;HYt7vF/$\"3-@L=8\\2;WF27$$\"3Y*******p(G**yF/$\"3ui(4H\"['**z%F27$$\"
3]mmmT6KU$)F/$\"3@aG$z[Z!3_F27$$\"3fKLLLbdQ()F/$\"37=W))[%[h]&F27$$\"3
6+]i!*z>W))F/$\"3))[L\")e5lmbF27$$\"3amm\"zW?)\\*)F/$\"3%ev#e6XT:cF27$
$\"3I+Dcw;j-!*F/$\"33ueDCJ]McF27$$\"3'HL3_!HWb!*F/$\"3K!*G%pqx%\\cF27$
$\"3rmT&Q8a#3\"*F/$\"3]Oa)3i4)fcF27$$\"3[++]i`1h\"*F/$\"3O+g&zqy[m&F27
$$\"3Z+](oW7;@*F/$\"3#\\i2J*[7kcF27$$\"3Y++DJ&f@E*F/$\"3#[-(p'37rl&F27
$$\"3Y+]i:mq7$*F/$\"3$\\I(4b;#Hk&F27$$\"3Y++++PDj$*F/$\"3#>XIQ+R/i&F27
$$\"3Y++voyMk%*F/$\"3Zbeb`2xWbF27$$\"3W++]P?Wl&*F/$\"3&\\rd[_6UT&F27$$
\"3K+]7G:3u'*F/$\"3Mt%4it;5=&F27$$\"3A++v=5s#y*F/$\"3iXAU#p:Dy%F27$$\"
3;+D1k2/P)*F/$\"3'QkvOzN?Z%F27$$\"35+]P40O\"*)*F/$\"31^-MS-*3-%F27$$\"
3k]7.#Q?&=**F/$\"3\"[&*GBGxFq$F27$$\"31+voa-oX**F/$\"3q-u_#Rm)pKF27$$
\"3ACc,\">g#f**F/$\"31a\\]%\\8*yHF27$$\"3[\\PMF,%G(**F/$\"35*phQ#[p'f#
F27$$\"3uu=nj+U')**F/$\"3\\A$G[2&*e-#F27$$\"\"\"F*F+-%'COLOURG6&%$RGBG
$\"#5F)F+F+-%+AXESLABELSG6$Q\"x6\"Q!Ffgl-%%VIEWG6$;F(Fjfl%(DEFAULTG" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "By comput
ing the derivative of  " }{XPPEDIT 18 0 "g(x)-phi[n](x);" "6#,&-%\"gG6
#%\"xG\"\"\"-&%$phiG6#%\"nG6#F'!\"\"" }{TEXT -1 50 ", we can find a fo
rmula for the positive value of " }{TEXT 275 1 "x" }{TEXT -1 37 " wher
e the maximum difference occurs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "Diff(g(x)-phi(n,x),x):\nh :
= unapply(normal(value(%)),n,x);\neq := algsubs(1-x^2=u,h(n,x)=0);\nuu
 := simplify(solve(eq,u));\nxmax := sqrt(1-uu);\nsimplify(g(xmax)-phi(
n,xmax),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6$%\"nG%
\"xG6\"6$%)operatorG%&arrowGF),$*,9%\"\"\",**&9$F0)F/\"\"#F0F0F3!\"\"*
(),&F0F0*$F4F0F6,$*&#F0F5F0*&,&F3F0F0F0F0F3F6F0F0F0F9F=F3F0F0*&F8F0F9F
=F0F0F9#F6F5F3F6,&F:F0F0F6F6F6F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#eqG/**%\"uG#!\"$\"\"#%\"nG!\"\"%\"xG\"\"\",(*()F',$*(F*F,,&F+F.F.
F.F.F+F,F.F.F'#F.F*F+F.F.*&F+F.F'F.F,*&F1F.F'F5F.F.\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#uuG)*&%\"nG\"\"#,&F'\"\"\"F*F*!\"#F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xmaxG*$,&\"\"\"F')*&%\"nG\"\"#,&F*F
'F'F'!\"#F*!\"\"#F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)%\"nGF&
\"\"\"),&F&F'F'F',$F&!\"\"F'F'*&)F&F)F')F),&F&F+F'F+F'F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The max
imum difference between  " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%
\"nG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "psi(x) = sqrt(1-x^2);
" "6#/-%$psiG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F'\"\"#!\"\"" }{TEXT -1 14 "
 occurs where " }{XPPEDIT 18 0 "abs(x) = sqrt(1-(n/(n+1))^(2*n));" "6#
/-%$absG6#%\"xG-%%sqrtG6#,&\"\"\"F,)*&%\"nGF,,&F/F,F,F,!\"\"*&\"\"#F,F
/F,F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "The correspondin
g difference is " }{XPPEDIT 18 0 "(n/(n+1))^n-(n/(n+1))^(n+1)" "6#,&)*
&%\"nG\"\"\",&F&F'F'F'!\"\"F&F')*&F&F',&F&F'F'F'F),&F&F'F'F'F)" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 8 "Now, as " }{XPPEDIT 18 0 "n ->infinity" "6#f*6#%\"nG7\"6$%
)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 4 ",   " }{XPPEDIT 
18 0 "((n/(n+1))^n) ->1/exp(1)" "6#f*6#)*&%\"nG\"\"\",&F'F(F(F(!\"\"F'
7\"6$%)operatorG%&arrowG6\"*&F(F(-%$expG6#F(F*F/F/F/" }{TEXT -1 7 "  a
nd  " }{XPPEDIT 18 0 "((n/(n+1))^(n+1)) ->1/exp(1)" "6#f*6#)*&%\"nG\"
\"\",&F'F(F(F(!\"\",&F'F(F(F(7\"6$%)operatorG%&arrowG6\"*&F(F(-%$expG6
#F(F*F0F0F0" }{TEXT -1 6 ",  so " }{XPPEDIT 18 0 "Limit((n/(n+1))^n-(n
/(n+1))^(n+1),n=infinity)=0" "6#/-%&LimitG6$,&)*&%\"nG\"\"\",&F*F+F+F+
!\"\"F*F+)*&F*F+,&F*F+F+F+F-,&F*F+F+F+F-/F*%)infinityG\"\"!" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 218 "Limit((n/(n+1))^n,n=infinity)=limit((n/(n+1))^n,n=in
finity);\nLimit((n/(n+1))^(n+1),n=infinity)=limit((n/(n+1))^n,n=infini
ty);\nLimit((n/(n+1))^n-(n/(n+1))^(n+1),n=infinity)=limit((n/(n+1))^n-
(n/(n+1))^(n+1),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Li
mitG6$)*&%\"nG\"\"\",&F)F*F*F*!\"\"F)/F)%)infinityG-%$expG6#F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$)*&%\"nG\"\"\",&F)F*F*F*!
\"\"F+/F)%)infinityG-%$expG6#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
&LimitG6$,&)*&%\"nG\"\"\",&F*F+F+F+!\"\"F*F+)F)F,F-/F*%)infinityG\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "epsi
lon = 10^(-6);" "6#/%(epsilonG)\"#5,$\"\"'!\"\"" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 12 "We can find " }{TEXT 276 1 "n" }{TEXT -1 
10 " so that  " }{XPPEDIT 18 0 "(n/(n+1))^n-(n/(n+1))^(n+1)=epsilon" "
6#/,&)*&%\"nG\"\"\",&F'F(F(F(!\"\"F'F()*&F'F(,&F'F(F(F(F*,&F'F(F(F(F*%
(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "fsolv
e((n/(n+1))^n-(n/(n+1))^(n+1)=10^(-6),n=40000);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+7%*yyO!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 277 1 "n" }{TEXT -1 56 " is \+
greater than 367878, the maximum difference between " }{XPPEDIT 18 0 "
phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "psi(x) = sqrt(1-x^2);" "6#/-%$psiG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$F
'\"\"#!\"\"" }{TEXT -1 21 " is always less than " }{XPPEDIT 18 0 "epsi
lon" "6#%(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "phi := (n,x) -> (1-x^2)^(1/2
+1/(2*n)):\ng := x -> sqrt(1-x^2):\nplot(g(x)-phi(367879,x),x=-1..1);
" }}{PARA 13 "" 1 "" {GLPLOT2D 389 146 146 {PLOTDATA 2 "6%-%'CURVESG6$
7]p7$$!\"\"\"\"!$F*F*7$$!3]LekynP')**!#=$\"3#QS$eG&>')=%!#C7$$!3-n;HdN
vs**F/$\"3)zqR%)*\\IF_F27$$!3_+v$fLI\"f**F/$\"3Q6E1Mn/.fF27$$!3/MLe9r]
X**F/$\"3o2!zN\"3E2kF27$$!3/,](=ng#=**F/$\"3^$3]z(Q5TrF27$$!3%pmm\"HU,
\"*)*F/$\"3_&3tn$QCnwF27$$!3()***\\PM@l$)*F/$\"3EyQ#4]w=R)F27$$!3!RLL$
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fF27$$!3[++](y$pZiF/$\"3w>jRwb[^_F27$$!33LLL$yaE\"eF/$\"33ru?!Qj(fXF27
$$!3hmmm\">s%HaF/$\"3'>(=qZgM')RF27$$!3Q+++]$*4)*\\F/$\"3/89=xMh$Q$F27
$$!39+++]_&\\c%F/$\"3*oht,cde#GF27$$!31+++]1aZTF/$\"3[&3iKtTXL#F27$$!3
umm;/#)[oPF/$\"3Yy3')=5IG>F27$$!3hLLL$=exJ$F/$\"3%ym8xBJ_\\\"F27$$!3*R
LLLtIf$HF/$\"3sEAoO?9r6F27$$!3]++]PYx\"\\#F/$\"3W+YA(p!RP%)!#D7$$!3EML
LL7i)4#F/$\"3ye`p1XW&)fF]v7$$!3c****\\P'psm\"F/$\"3[r]V5I+yPF]v7$$!3')
****\\74_c7F/$\"3/S'*[tC&e9#F]v7$$!3)3LLL3x%z#)!#>$\"3w#)3#*)\\ooJ*!#E
7$$!3KMLL3s$QM%F`w$\"3Qz%=Q7bXc#Fcw7$$!3]^omm;zr)*!#@$\"3w6yPog\\C8!#H
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01E!*Fcw7$$\"3GKLLe\"*[H7F/$\"3/D%>DB>X0#F]v7$$\"3I*******pvxl\"F/$\"3
![c&*3;-^t$F]v7$$\"3#z****\\_qn2#F/$\"3!y+o$GSZheF]v7$$\"3U)***\\i&p@[
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mZvOLF/$\"3,Q'[k\"=Q7:F27$$\"3i******\\2goPF/$\"3wz#*e`fTG>F27$$\"3UKL
$eR<*fTF/$\"3sP(yDe\\%[BF27$$\"3m******\\)Hxe%F/$\"3)G>&34r(R&GF27$$\"
3ckm;H!o-*\\F/$\"3'H)>#=U-JP$F27$$\"3y)***\\7k.6aF/$\"3G(>*3$)[kfRF27$
$\"3#emmmT9C#eF/$\"3mfJg5'=[d%F27$$\"33****\\i!*3`iF/$\"3%oNT&f+Kg_F27
$$\"3%QLLL$*zym'F/$\"3uE=[!HCm&fF27$$\"3wKLL3N1#4(F/$\"3_mimKy'))p'F27
$$\"3Nmm;HYt7vF/$\"3g%*z*[GoZX(F27$$\"3Y*******p(G**yF/$\"3=Am6_6*>:)F
27$$\"3]mmmT6KU$)F/$\"3G(\\!>jX\\A*)F27$$\"3fKLLLbdQ()F/$\"3$>r-Bnn3`*
F27$$\"3amm\"zW?)\\*)F/$\"31/5Fo+X)y*F27$$\"3[++]i`1h\"*F/$\"3=e%*G+$)
yg**F27$$\"3Z+](oW7;@*F/$\"3q@Cq1$\\N)**F27$$\"3Y++DJ&f@E*F/$\"3IU%4%
\\\\&p***F27$$\"3Y+]i:mq7$*F/$\"3sFU\\/3`****F27$$\"3Y++++PDj$*F/$\"37
<D\\rlZ*)**F27$$\"3X+]P%y+QT*F/$\"3CFdo&>jX'**F27$$\"3Y++voyMk%*F/$\"3
D=xNEd)>#**F27$$\"3W+]7`\\*[^*F/$\"3g>\\QBn9e)*F27$$\"3W++]P?Wl&*F/$\"
3cN<&*RkKo(*F27$$\"3K+]7G:3u'*F/$\"3C]U5qJ6a%*F27$$\"3A++v=5s#y*F/$\"3
sl$4'\\*\\v'))F27$$\"3;+D1k2/P)*F/$\"3oUDhdkN'Q)F27$$\"35+]P40O\"*)*F/
$\"3O'>m%*oV9m(F27$$\"3k]7.#Q?&=**F/$\"3r)3bP@%GNrF27$$\"31+voa-oX**F/
$\"36JGI09f,kF27$$\"3ACc,\">g#f**F/$\"3Cu**[bGd(*eF27$$\"3[\\PMF,%G(**
F/$\"3H;S[Nm=A_F27$$\"3uu=nj+U')**F/$\"3C>p(RK;U=%F27$$\"\"\"F*F+-%'CO
LOURG6&%$RGBG$\"#5F)F+F+-%+AXESLABELSG6$Q\"x6\"Q!Feel-%%VIEWG6$;F(Fidl
%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "From this analysis w
e see that the sequence of functions " }{XPPEDIT 18 0 "phi[n](x);" "6#
-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 24 " converges uniformly to " }
{XPPEDIT 18 0 "g(x) = sqrt(1-x^2);" "6#/-%\"gG6#%\"xG-%%sqrtG6#,&\"\"
\"F,*$F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 31 "It is interesting to note that " }
{XPPEDIT 18 0 "(sqrt(1-(n/(n+1))^(2*n))) ->sqrt(1-1/exp(2)) " "6#f*6#-
%%sqrtG6#,&\"\"\"F))*&%\"nGF),&F,F)F)F)!\"\"*&\"\"#F)F,F)F.7\"6$%)oper
atorG%&arrowG6\"-F&6#,&F)F)*&F)F)-%$expG6#F0F.F.F5F5F5" }{TEXT -1 5 " \+
 as " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arr
owG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Limit(sqrt(1-(n/(n+1))^
(2*n)),n=infinity);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%&LimitG6$*$-%%sqrtG6#,&\"\"\"F+)*&%\"nGF+,&F.F+F+F+!
\"\",$F.\"\"#F0F+/F.%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-
%$expG6#!\"\"\"\"\"-%%sqrtG6#,&-F%6#\"\"#F(F'F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+]\\t)H*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }{XPPEDIT 18 0 "Int(phi[n]
(x),x = -1 .. 1);" "6#-%$IntG6$-&%$phiG6#%\"nG6#%\"xG/F,;,$\"\"\"!\"\"
F0" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n=1,2,3,` . . . `" "6&/%\"nG\"\"
\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 15 ", converges to " }{XPPEDIT 18 0 
"Int(sqrt(1-x^2),x = -1 .. 1) = Pi/2;" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"
F+*$%\"xG\"\"#!\"\"/F-;,$F+F/F+*&%#PiGF+F.F/" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
88 "phi := (n,x) -> (1-x^2)^(1/2+1/(2*n)):\na := [seq(evalf(Int(phi(n,
x),x=-1..1)),n=1..15)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"aG71$\"
+LLLL8!\"*$\"+#GoxV\"F($\"+?$[$y9F($\"+v([**\\\"F($\"+$okL^\"F($\"+>x]
A:F($\"+e!R\"H:F($\"+.)oT`\"F($\"+2Z6Q:F($\"+\\IHT:F($\"+^z!Ra\"F($\"+
Yq4Y:F($\"+:l&za\"F($\"+*fb&\\:F($\"+8a%4b\"F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot([Pi/2,
[seq([n,a[n]],n=1..15)]$2],0..15,symbol=circle,\n   style=[line$2,poin
t],linestyle=[1,2],color=[cyan,grey,blue]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 376 204 204 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$\"\"!F)$\"3/+++F
jzq:!#<7$$\"3')*****\\7t&pK!#=F*7$$\"3$****\\(=7T9hF0F*7$$\"3X****\\(=
HPJ*F0F*7$$\"3;++DJaU`7F,F*7$$\"3)***\\P%GZRd\"F,F*7$$\"3%)**\\(=276(=
F,F*7$$\"3'***\\(o**3)y@F,F*7$$\"3/+](ofHq\\#F,F*7$$\"3.+]Pf'HU\"GF,F*
7$$\"33++]7*309$F,F*7$$\"3:++Dce*yU$F,F*7$$\"3;++]([D9v$F,F*7$$\"3c***
*\\iNGwSF,F*7$$\"37++]7XM*Q%F,F*7$$\"3/+](o%QjtYF,F*7$$\"32++]i8o6]F,F
*7$$\"3i******\\>0)H&F,F*7$$\"3Y**\\(=-p6j&F,F*7$$\"3d*****\\2Mg#fF,F*
7$$\"35+](=xZ&\\iF,F*7$$\"3;+]i:$4wb'F,F*7$$\"3-++v=#R!zoF,F*7$$\"3q+]
P4A@urF,F*7$$\"3I++Dchf#\\(F,F*7$$\"3))**\\(of2L#yF,F*7$$\"3M**\\7yG>6
\")F,F*7$$\"3w++voo6A%)F,F*7$$\"3q*****\\xJLu)F,F*7$$\"3W++v$*ydd!*F,F
*7$$\"3#***\\(=<F;O*F,F*7$$\"35***\\i0A#*p*F,F*7$$\"3*)****\\2mD+5!#;F
*7$$\"3*****\\i0XE.\"FhqF*7$$\"3%**\\(o/Q*>1\"FhqF*7$$\"3=++vQ(zS4\"Fh
qF*7$$\"3***\\(=-,FC6FhqF*7$$\"33+v$4tFe:\"FhqF*7$$\"3!****\\73\"o'=\"
FhqF*7$$\"3-+voz;)*=7FhqF*7$$\"31+++&*44]7FhqF*7$$\"35+]7jZ!>G\"FhqF*7
$$\"34+v=(4bMJ\"FhqF*7$$\"3;++]xlWU8FhqF*7$$\"39+]i&3ucP\"FhqF*7$$\"3
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oxV\"F,7$$\"\"$F)$\"31+++?$[$y9F,7$$\"\"%F)$\"3-+++v([**\\\"F,7$$\"\"&
F)$\"3/+++$okL^\"F,7$$\"\"'F)$\"3/+++>x]A:F,7$$\"\"(F)$\"3%******z0R\"
H:F,7$$\"\")F)$\"3.+++.)oT`\"F,7$$\"\"*F)$\"33+++2Z6Q:F,7$$\"#5F)$\"33
+++\\IHT:F,7$$\"#6F)$\"3++++^z!Ra\"F,7$$\"#7F)$\"33+++Yq4Y:F,7$$\"#8F)
$\"33+++:l&za\"F,7$$\"#9F)$\"31+++*fb&\\:F,7$Fgt$\"3++++8a%4b\"F,-Fjt6
&F\\u$\")=THvF_uFezFezF`u-Feu6#Fav-F$6&Fju-Fjt6&F\\uF(F(F]u-Fau6#%&POI
NTGFdu-%+AXESLABELSG6$Q!6\"Fc[l-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F(Fgt%(
DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the sequence of functions \+
" }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~
G" }{TEXT -1 48 ", defined on the set |R of all real numbers by: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x) = PIECEWISE
([1/(1+(x-n)^2), x < n],[1, n <= x]);" "6#/-&%$phiG6#%\"nG6#%\"xG-%*PI
ECEWISEG6$7$*&\"\"\"F0,&F0F0*$,&F*F0F(!\"\"\"\"#F0F42F*F(7$F01F(F*" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "The sequence " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"
\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 12 ", converges " }{TEXT 261 9 "poin
twise" }{TEXT -1 17 " to the function " }{XPPEDIT 18 0 "g(x) = 0;" "6#
/-%\"gG6#%\"xG\"\"!" }{TEXT -1 16 " on the set |R. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "phi := (n
,x) -> piecewise(x<n,1/(1+(x-n)^2),1):\n'phi(n,x)'=phi(n,x);\nplot([ph
i(4,x),phi(3,x),phi(2,x),phi(1,x)],x=-5..5,\n      color=[red,blue,gre
en,magenta]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6$%\"nG%\"xG-
%*PIECEWISEG6$7$*&\"\"\"F.,&F.F.*$),&F(F.F'!\"\"\"\"#F.F.F32F(F'7$F.%*
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F*4W/_h&**F[r7$$\"3=-]7GtB)f*F[r$\"3gwqz(o%)Q)**F[r7$$\"3W-+v$44,')*F[
r$\"3C6RKOM/)***F[r7$$\"3Q+v$f3)>75F1Fc[l7$FhtFc[l7$F^^mFc[l7$F]uFc[l7
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Q!F\\[n-%%VIEWG6$;F(Fb\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "For examp
le, taking " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 15 ", t
he sequence " }{XPPEDIT 18 0 "phi[n](2);" "6#-&%$phiG6#%\"nG6#\"\"#" }
{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 171 "phi := (n,x) -> piecewise(x<n,1/(1+(x-n)^2),1):\nplo
t([1/(1+(2-x)^2),[seq([n,phi(n,2)],n=1..10)]],x=2..13,\n symbol=circle
,style=[line,point],linestyle=2,color=[grey,blue]);" }}{PARA 13 "" 1 "
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Fb^l-%&STYLEG6#%%LINEG-F$6%7,7$F+F+F'7$$\"\"$F*$\"3++++++++]F37$$\"\"%
F*$\"35+++++++?F37$$\"\"&F*$\"3/+++++++5F37$$\"\"'F*$\"3_qk<THN#)eFas7
$$\"\"(F*$\"3QYQ:YQ:YQFas7$$\"\")F*$\"3&Gq-Fq-Fq#Fas7$$\"\"*F*$\"3/+++
++++?Fas7$$\"#5F*$\"3bQ:YQ:YQ:Fas-F_^l6&Fa^l$F*F*Fgal$\"*++++\"Fd^l-Ff
^l6#%&POINTG-%*LINESTYLEG6#F)-%+AXESLABELSG6$Q\"x6\"Q!Fdbl-%'SYMBOLG6#
%'CIRCLEG-%%VIEWG6$;F(Fj]l%(DEFAULTG" 1 2 4 2 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sequence of functions  " }
{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 6 " do
es " }{TEXT 260 3 "not" }{TEXT -1 10 " converge " }{TEXT 261 9 "unifor
mly" }{TEXT -1 5 " to  " }{XPPEDIT 18 0 "g(x) = 0;" "6#/-%\"gG6#%\"xG
\"\"!" }{TEXT -1 16 " on the set |R. " }}{PARA 0 "" 0 "" {TEXT -1 31 "
The maximum difference between " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$ph
iG6#%\"nG6#%\"xG" }{TEXT -1 5 ", as " }{TEXT 278 1 "x" }{TEXT -1 24 " \+
varies, and 0 is 1. If " }{XPPEDIT 18 0 "epsilon< 1" "6#2%(epsilonG\"
\"\"" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "epsilon < abs(phi[n](x));
" "6#2%(epsilonG-%$absG6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 9 " for any
 " }{TEXT 279 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x>=n" "6#1%\"
nG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{PARA 0 "" 0 "" {TEXT -1 32 "Consider the sequence functions " }
{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }
{TEXT -1 26 ", defined on the interval " }{XPPEDIT 18 0 "[0,2]" "6#7$
\"\"!\"\"#" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "phi[n](x) = 1/(1+n^3*(x-1/n)^2);" "6#/-&%$phiG6#%\"nG6#
%\"xG*&\"\"\"F,,&F,F,*&F(\"\"$,&F*F,*&F,F,F(!\"\"F2\"\"#F,F2" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }{XPPEDIT 18 0 
"phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 1,2,
3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 12 ", conv
erges " }{TEXT 261 9 "pointwise" }{TEXT -1 17 " to the function " }
{XPPEDIT 18 0 "g(x) = 0;" "6#/-%\"gG6#%\"xG\"\"!" }{TEXT -1 8 " on the
 " }{XPPEDIT 18 0 "[0,2" "6#7$\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "phi \+
:= (n,x) -> 1/(1+n^3*(x-1/n)^2);\nplot([phi(1,x),phi(2,x),phi(4,x),phi
(8,x),phi(16,x),phi(32,x)],x=0..2,\n   color=[red,blue,green,magenta,c
oral,cyan]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGf*6$%\"nG%\"xG6
\"6$%)operatorG%&arrowGF)*&\"\"\"F.,&F.F.*&)9$\"\"$F.),&9%F.*&F.F.F2!
\"\"F8\"\"#F.F.F8F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 529 270 270 
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 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Even though the \"hump
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iG6#%\"nG6#%\"xG" }{TEXT -1 21 " never disappears as " }{XPPEDIT 18 0 
"n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*
" }{TEXT -1 108 ", it becomes progressively narrower at a rate which i
s fast enough to ensure that the sequence of functions " }{XPPEDIT 18 
0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 26 " converges poi
ntwise to 0." }}{PARA 0 "" 0 "" {TEXT -1 66 "The following picture ill
ustrates the convergence of the sequence " }{XPPEDIT 18 0 "phi[n](1/10
);" "6#-&%$phiG6#%\"nG6#*&\"\"\"F*\"#5!\"\"" }{TEXT -1 6 " to 0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
162 "phi := (n,x) -> 1/(1+n^3*(x-1/n)^2):\nplot([phi(x,1/10),[seq([n,p
hi(n,1/10)],n=1..30)]],x=0..30,\n  symbol=circle,style=[line,point],li
nestyle=2,color=[grey,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 
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}}}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "x=0" "6#/%\"xG
\"\"!" }{TEXT -1 9 " we have " }{XPPEDIT 18 0 "phi[n](0) = 1/(1+n);" "
6#/-&%$phiG6#%\"nG6#\"\"!*&\"\"\"F,,&F,F,F(F,!\"\"" }{TEXT -1 26 ", wh
ich converges to 0 as " }{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$
%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 13 "Suppose that " }{TEXT 280 1 "x" }{TEXT -1 13 " is po
sitive." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "epsilon" 
"6#%(epsilonG" }{TEXT -1 60 " be a (small) positive number, and consid
er the inequality: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "abs(phi[n](x))<epsilon" "6#2-%$absG6#-&%$phiG6#%\"nG6#%\"xG%(epsilo
nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(1+n^3*(x-1/n)^2)<epsilon
" "6#2*&\"\"\"F%,&F%F%*&%\"nG\"\"$,&%\"xGF%*&F%F%F(!\"\"F-\"\"#F%F-%(e
psilonG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "This inequali
ty is equivalent to" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "1+n^3*(x-1/n)^2>1/epsilon" "6#2*&\"\"\"F%%(epsilonG!\"\",&F%F%*&%\"
nG\"\"$,&%\"xGF%*&F%F%F*F'F'\"\"#F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 7 "Choose " }{TEXT 281 1 "N" }{TEXT -1 10 " so that  " }
{XPPEDIT 18 0 "2/x < N;" "6#2*&\"\"#\"\"\"%\"xG!\"\"%\"NG" }{TEXT -1 
6 "  and " }{XPPEDIT 18 0 "1/epsilon < N+1;" "6#2*&\"\"\"F%%(epsilonG!
\"\",&%\"NGF%F%F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The
n for every " }{TEXT 282 1 "n" }{TEXT -1 11 " such that " }{XPPEDIT 
18 0 "n>=N" "6#1%\"NG%\"nG" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "
x - 1/n>1/n" "6#2*&\"\"\"F%%\"nG!\"\",&%\"xGF%*&F%F%F&F'F'" }{TEXT -1 
7 "  and  " }{XPPEDIT 18 0 "n^3*(x-1/n)^2>n" "6#2%\"nG*&F$\"\"$,&%\"xG
\"\"\"*&F)F)F$!\"\"F+\"\"#" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "
1+n^3*(x-1/n)^2>1/epsilon" "6#2*&\"\"\"F%%(epsilonG!\"\",&F%F%*&%\"nG
\"\"$,&%\"xGF%*&F%F%F*F'F'\"\"#F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "This means that  " }
{XPPEDIT 18 0 "abs(phi[n](x)) = phi[n](x);" "6#/-%$absG6#-&%$phiG6#%\"
nG6#%\"xG-&F)6#F+6#F-" }{XPPEDIT 18 0 " ``< epsilon" "6#2%!G%(epsilonG
" }{TEXT -1 43 ", and shows that the sequence of functions " }
{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 26 " c
onverges pointwise to 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "x = 2*`.`*10^(-6
);" "6#/%\"xG*(\"\"#\"\"\"%\".GF')\"#5,$\"\"'!\"\"F'" }{TEXT -1 5 " an
d " }{XPPEDIT 18 0 "epsilon=10^(-6)" "6#/%(epsilonG)\"#5,$\"\"'!\"\"" 
}{TEXT -1 6 ", let " }{XPPEDIT 18 0 "N=10^6+1" "6#/%\"NG,&*$\"#5\"\"'
\"\"\"F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xx := 2*10^(-6);\nevalf(phi(10^6+1,
xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG#\"\"\"\"'++]" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+++%*****!#;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The maximum point on th
e graph of " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }
{TEXT -1 14 " occurs where " }{XPPEDIT 18 0 "x=1/n" "6#/%\"xG*&\"\"\"F
&%\"nG!\"\"" }{TEXT -1 36 " , and the maximum value there is 1." }}
{PARA 0 "" 0 "" {TEXT -1 42 "It follows that the sequence of functions
 " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~
G" }{TEXT -1 7 ", does " }{TEXT 260 3 "not" }{TEXT -1 10 " converge " 
}{TEXT 261 9 "uniformly" }{TEXT -1 6 " to 0." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "Int(1/(1+n^
3*(x-1/n)^2),x = 0 .. 1) = (arctan(sqrt(n)*(n-1))+arctan(sqrt(n)))/(n^
(3/2));" "6#/-%$IntG6$*&\"\"\"F(,&F(F(*&%\"nG\"\"$,&%\"xGF(*&F(F(F+!\"
\"F0\"\"#F(F0/F.;\"\"!F(*&,&-%'arctanG6#*&-%%sqrtG6#F+F(,&F+F(F(F0F(F(
-F86#-F<6#F+F(F()F+*&F,F(F1F0F0" }{TEXT -1 23 ",  which tends to 0 as \+
" }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6
\"%)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Int(1/(1+n^3*(x-1/n)^2),x=0
..1);\nvalue(%);\nLimit(%,n=infinity);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&F'F'*&)%\"nG\"\"$F'),&%\"xGF'*&F'F
'F+!\"\"F1\"\"#F'F'F1/F/;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&
,&-%'arctanG6#*&-%%sqrtG6#%\"nG\"\"\",&F,F-!\"\"F-F-F--F&6#*$F)F-F-F-*
$)F,#\"\"$\"\"#F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&
-%'arctanG6#*&-%%sqrtG6#%\"nG\"\"\",&F/F0!\"\"F0F0F0-F)6#*$F,F0F0F0*$)
F/#\"\"$\"\"#F0F2/F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 35 "Consider the sequence of functions " }{XPPEDIT 18 0 "phi[
n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` .
 . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 26 ", defined o
n the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT 
-1 5 " by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n]
(x) = PIECEWISE([1-1/(2*n)-n*x^2/2, abs(x) < 1/n],[1-abs(x), 1/n <= ab
s(x)]);" "6#/-&%$phiG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$,(\"\"\"F0*&F0F0*&
\"\"#F0F(F0!\"\"F4*(F(F0*$F*F3F0F3F4F42-%$absG6#F**&F0F0F(F47$,&F0F0-F
96#F*F41*&F0F0F(F4-F96#F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " The sequence " }{XPPEDIT 18 0 "ph
i[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,`
 . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 38 ", converg
es pointwise to the function " }{XPPEDIT 18 0 "g(x) = 1-abs(x);" "6#/-
%\"gG6#%\"xG,&\"\"\"F)-%$absG6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "phi :=
 (n,x) -> piecewise(abs(x)<1/n,1-1/(2*n)-n*x^2/2,1-abs(x)):\n'phi(n,x)
'=phi(n,x);\nplot([1-abs(x),phi(1,x),phi(2,x),phi(3,x),phi(4,x),phi(5,
x),phi(6,x)],x=-1..1,\ncolor=[black,red,blue,green,magenta,coral,cyan]
,linestyle=[2,1$6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6$%\"n
G%\"xG-%*PIECEWISEG6$7$,(\"\"\"F.*&F.F.*&\"\"#F.F'F.!\"\"F2*(F1F2F'F.F
(F1F22-%$absG6#F(*&F.F.F'F27$,&F.F.F5F2%*otherwiseG" }}{PARA 13 "" 1 "
" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6%7U7$$!\"\"\"\"!$F*F
*7$$!3ommm;p0k&*!#=$\"39LLLL3VfV!#>7$$!3wKL$3<XZ=*F/$\"3]smm\"H[D:)F27
$$!3mmmmT%p\"e()F/$\"3LLLLe0$=C\"F/7$$!3:mmm\"4m(G$)F/$\"3'QLL$3RBr;F/
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$$!3u****\\P8#\\4(F/$\"3C++]i'y]!HF/7$$!3+nm;/siqmF/$\"3,LL$ezs$HLF/7$
$!3[++](y$pZiF/$\"3_****\\7iI_PF/7$$!33LLL$yaE\"eF/$\"3#pmmm@Xt=%F/7$$
!3hmmm\">s%HaF/$\"3QLLL3y_qXF/7$$!3Q+++]$*4)*\\F/$\"3i******\\1!>+&F/7
$$!39+++]_&\\c%F/$\"3()******\\Z/NaF/7$$!31+++]1aZTF/$\"3'*******\\$fC
&eF/7$$!3umm;/#)[oPF/$\"3ELL$ez6:B'F/7$$!3hLLL$=exJ$F/$\"3Smmm;=C#o'F/
7$$!3*RLLLtIf$HF/$\"3-mmmm#pS1(F/7$$!3]++]PYx\"\\#F/$\"3]****\\i`A3vF/
7$$!3EMLLL7i)4#F/$\"3slmmm(y8!zF/7$$!3c****\\P'psm\"F/$\"3V++]i.tK$)F/
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7$$\"3%pJL$ezw5VF2$\"3Iom;/K#*o&*F/7$$\"3s*)***\\PQ#\\\")F2$\"3-,+]ih2
&=*F/7$$\"3GKLLe\"*[H7F/$\"3snmmT3^q()F/7$$\"3I*******pvxl\"F/$\"3q+++
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nm;/E3SeF/7$$\"3m******\\)Hxe%F/$\"3M+++],F7aF/7$$\"3ckm;H!o-*\\F/$\"3
VNL$3(>t4]F/7$$\"3y)***\\7k.6aF/$\"3?,+](ej*)e%F/7$$\"3#emmmT9C#eF/$\"
3=MLL$e&exTF/7$$\"33****\\i!*3`iF/$\"3#4++v$4\"pu$F/7$$\"3%QLLL$*zym'F
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\"3mLL$3Pls[#F/7$$\"3Y*******p(G**yF/$\"3`++++Br+@F/7$$\"3]mmmT6KU$)F/
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/7$F\\t$\"3Y.Z!)R'32*\\F/7$Fat$\"3)Rr`p&\\zm\\F/7$Fft$\"3ZRqZ?yTC\\F/7
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\"3Q!)pU')>3qXF/7$F_v$\"3QkjN\"Q.LW%F/7$Fdv$\"3Gs\\N>C))*G%F/7$Fiv$\"3
Eh@*HOaZ8%F/7$F^w$\"3)*[4<TnjZRF/7$Fcw$\"3]_Q&)\\7'[v$F/7$Fhw$\"3=d*Hr
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F,F3F8F=FBFGFLFQFVFenFjn7$F`o$\"3MdHv)G+>+&F/7$Feo$\"3%Ru\\lN=hT&F/7$F
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F/7$Fft$\"3%*yS&4kN)[tF/7$F[u$\"3p]*[G(z<DsF/7$F`u$\"3=B\"\\'=CqoqF/7$
Feu$\"3;d)*HEM)Q)oF/7$Fju$\"3wgR&G(R;SmF/7$F_v$\"3wGFring'Q'F/7$Fdv$\"
3aW*4(Q[wzgF/7$Fiv$\"3_AV)fs3&pdF/7$F^w$\"3'z*=M#[t_R&F/7$Fcw$\"3+0xq*
\\A(4]F/FgwF\\xFaxFfxF[yF`yFeyFjyF_zFdzFizF^[l-Fb[l6&Fd[lF+F+F[elF^el-
F$6%7SF'F,F3F8F=FBFGFLFQFVFenFjnF_oFdoFioF^p7$Fdp$\"3%*Q?()Ga?#o'F/7$F
ip$\"33`OpU*z./(F/7$F^q$\"3&\\4?n?#*>S(F/7$Fcq$\"3#[q$=r;qswF/7$Fhq$\"
3yERrE^O;zF/7$F]r$\"3#*e3!Gh1l4)F/7$Fbr$\"3)4<t@s30B)F/7$Fgr$\"3R#e%z]
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)F/7$Fft$\"3@`Ww%z'e1\")F/7$F[u$\"3&3w1EH+6#zF/7$F`u$\"3/>qIhpQ'o(F/7$
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[yF`yFeyFjyF_zFdzFizF^[l-Fb[l6&Fd[lF+F[elF+F^el-F$6%7WF'F,F3F8F=FBFGFL
FQFVFenFjnF_oFdoFioF^pFcpFhp7$F^q$\"3p[B=J=@3vF/7$Fcq$\"3e&\\+Qyd\"pyF
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gKL$e9d;J'F2$\"3[-p9ofKq')F/7$Fgr$\"3E)*\\hc@E7()F/7$F\\s$\"36o1)>&=8S
()F/7$Fas$\"3Eeea40)*\\()F/7$Fgs$\"3\\[KA<$H6u)F/7$F\\t$\"3*Q\")=#fX$G
r)F/7$$\"3L`mmmJ+IiF2$\"3uYm3@TPs')F/7$Fat$\"3\"[&[\"y#)zrh)F/7$Fft$\"
3'*e\"3>GrwW)F/7$F[u$\"3\\-zpXfN+#)F/7$F`u$\"3OY#)HP[S()yF/7$Feu$\"3A8
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()*)F/7$Fas$\"3]AB$pjv****)F/7$Fgs$\"3sf!Hlk6*))*)F/7$F\\t$\"3-<N-*>VN
&*)F/7$F\\`m$\"3#*3$e8lnH!*)F/7$Fat$\"3,p&oZyuR$))F/7$Fft$\"3l)>&Q-\"*
3A')F/7$F[u$\"31yB7K\\%HJ)F/F_uFduFiuF^vFcvFhvF]wFbwFgwF\\xFaxFfxF[yF`
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$\"3YowjWW'=:*F/7$Fas$\"3q`a)4VPm;*F/7$Fgs$\"3#z`,DkgL:*F/7$F\\t$\"38(
)[\\0&=46*F/7$F\\`m$\"3'pj'H[yA]!*F/7$Fat$\"31\\*)Q3kVn*)F/7$Fft$\"3?/
*G&*etJr)F/7$F[u$\"3[>N@&e+AM)F/F_uFduFiuF^vFcvFhvF]wFbwFgwF\\xFaxFfxF
[yF`yFeyFjyF_zFdzFizF^[l-Fb[l6&Fd[lF+F[elF[elF^el-%+AXESLABELSG6$Q\"x6
\"Q!F\\gm-%%VIEWG6$;F(F_[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 32 "The maximum difference between  " }
{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 5 " an
d " }{XPPEDIT 18 0 "g(x) = 1-abs(x);" "6#/-%\"gG6#%\"xG,&\"\"\"F)-%$ab
sG6#F'!\"\"" }{TEXT -1 6 "  is  " }{XPPEDIT 18 0 "1/(2*n);" "6#*&\"\"
\"F$*&\"\"#F$%\"nGF$!\"\"" }{TEXT -1 18 ", and occurs when " }
{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 29 "Since the maximum difference " }{XPPEDIT 18 0 "1/(2*n)
" "6#*&\"\"\"F$*&\"\"#F$%\"nGF$!\"\"" }{TEXT -1 19 "  tends to zero as
 " }{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG
6\"%)infinityGF*F*F*" }{TEXT -1 44 ", it follows that the sequence of \+
functions " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ",
 " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~
.~.~G" }{TEXT -1 42 ", converges uniformly to the the function " }
{XPPEDIT 18 0 "g(x)=1-abs(x)" "6#/-%\"gG6#%\"xG,&\"\"\"F)-%$absG6#F'!
\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 24 "Note that each function " }{XPPEDIT 18 0 "phi[n];" "6#
&%$phiG6#%\"nG" }{TEXT -1 88 " is continuously differentiable, but the
 second derivative does not exist at the points " }{XPPEDIT 18 0 "x=-1
/n" "6#/%\"xG,$*&\"\"\"F'%\"nG!\"\"F)" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "x=1/n" "6#/%\"xG*&\"\"\"F&%\"nG!\"\"" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "psi[n]" "6#&%$
psiG6#%\"nG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%
\"nG" }{TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "psi[n](x)=``" "6#/-&%$psiG6#%\"nG6#%\"xG%!G" }
{XPPEDIT 18 0 "phi[n]*`'`(x) = PIECEWISE([1, x <= -1/n],[-n*x, -1/n < \+
x and x < 1/n],[-1, 1/n <= x]);" "6#/*&&%$phiG6#%\"nG\"\"\"-%\"'G6#%\"
xGF)-%*PIECEWISEG6%7$F)1F-,$*&F)F)F(!\"\"F57$,$*&F(F)F-F)F532,$*&F)F)F
(F5F5F-2F-*&F)F)F(F57$,$F)F51*&F)F)F(F5F-" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "psi
 := (n,x) -> piecewise(abs(x)<1/n,n*x,signum(x)):\n'psi(x,n)'=psi(n,x)
;\nplot([psi(1,x),psi(2,x),psi(4,x),signum(x)],x=-2..2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%$psiG6$%\"xG%\"nG-%*PIECEWISEG6$7$*&F(\"\"\"F
'F.2-%$absG6#F'*&F.F.F(!\"\"7$-%'signumGF2%*otherwiseG" }}{PARA 13 "" 
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WG6$;F(Fgu%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }{XPPEDIT 18 0 "Int(phi[
n](x),x = -1 .. 1);" "6#-%$IntG6$-&%$phiG6#%\"nG6#%\"xG/F,;,$\"\"\"!\"
\"F0" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n=1,2,3,` . . . `" "6&/%\"nG\"
\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 15 ", converges to " }{XPPEDIT 18 
0 "Int(``(1-abs(x)),x = -1 .. 1) = 1;" "6#/-%$IntG6$-%!G6#,&\"\"\"F+-%
$absG6#%\"xG!\"\"/F/;,$F+F0F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 9 "In fact  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(phi[n](x),x = -1 .. 1) = Int(``(1+x),x = -1 .. 1/n)+Int(``(1
-1/(2*n)-n*x^2/2),x = -1/n .. 1/n)+Int(``(1-x),x = 1/n .. 1);" "6#/-%$
IntG6$-&%$phiG6#%\"nG6#%\"xG/F-;,$\"\"\"!\"\"F1,(-F%6$-%!G6#,&F1F1F-F1
/F-;,$F1F2*&F1F1F+F2F1-F%6$-F76#,(F1F1*&F1F1*&\"\"#F1F+F1F2F2*(F+F1*$F
-FEF1FEF2F2/F-;,$*&F1F1F+F2F2*&F1F1F+F2F1-F%6$-F76#,&F1F1F-F2/F-;*&F1F
1F+F2F1F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = (3*n^2-1)/(3*n^2);" "6#/%!G*&,&*&\"\"$\"\"\"*$%\"n
G\"\"#F)F)F)!\"\"F)*&F(F)*$F+F,F)F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "``(Int(1+
x,x=-1..-1/n))+``(Int(1-1/(2*n)-n*x^2/2,x=-1/n..1/n))+\n  ``(Int(1-x,x
=1/n..1));\n``=map(value,%);\n``=eval(subs(``=(_u->_u),rhs(%)));\n``=n
ormal(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%!G6#-%$IntG6$,&%
\"xG\"\"\"F,F,/F+;!\"\",$*&F,F,%\"nGF/F/F,-F%6#-F(6$,(F,F,*&F,F,*&\"\"
#F,F2F,F/F/*(F:F/F2F,F+F:F//F+;F0F1F,-F%6#-F(6$,&F,F,F+F//F+;F1F,F," }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#\"\"\"-F$6#,(*&F(F(*&F'F
()%\"nGF'F(!\"\"F(#F(F'F(*&F(F(F/F0F0F(F(-F$6#,&*&F'F(F/F0F(*(\"\"%F(
\"\"$F0F/!\"#F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"\"F'*
&\"\"$F')%\"nG\"\"#F'!\"\"F-F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
!G,$*(\"\"$!\"\",&\"\"\"F(*&F'F*)%\"nG\"\"#F*F*F*F-!\"#F*" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Consider th
e sequence of functions " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" 
}{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"
\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 26 ", defined on the interval " }
{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 4 " by " }{XPPEDIT 
18 0 "phi[n](x) = x^n;" "6#/-&%$phiG6#%\"nG6#%\"xG)F*F(" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 14 " The sequence " }{XPPEDIT 18 0 "phi[
n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` .
 . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 12 ", converges
 " }{TEXT 261 9 "pointwise" }{TEXT -1 17 " to the function " }
{XPPEDIT 18 0 "g(x) = PIECEWISE([0, x < 1],[1, x = 1]);" "6#/-%\"gG6#%
\"xG-%*PIECEWISEG6$7$\"\"!2F'\"\"\"7$F./F'F." }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 23 "Note that the function " }{XPPEDIT 18 0 "g(x)" 
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18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "phi := (n,x) -> x
^n;\nplot([phi(1,x),phi(2,x),phi(4,x),phi(8,x),phi(16,x),phi(32,x)],x=
0..1,\n   color=[red,blue,green,magenta,coral,cyan]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$phiGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF))9%
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hs$\"3\"[()*\\p\"))[+&F:7$$\"3y***\\(oTAq#*F:$\"3#*z'pY%[3aaF:7$F[t$\"
3E2Ayk*p\"QfF:7$$\"3sK$eRA5\\Z*F:$\"3M6MFZTK&\\'F:7$F^t$\"3q5\"z+q$p(4
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Fat$\"3I([G'\\x=^\\F:7$$\"3v]iS\"*3))4)*F:$\"3#Gr%>Jh_5aF:7$Fb]n$\"3SU
<z*yG5\"fF:7$$\"3e\\(=nj+U')*F:$\"3)[BcF^biX'F:7$F[bm$\"3k(>\\#>G1]qF:
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,%G(**F:$\"39iA.(\\(\\m\"*F:Fct-Fgt6&FitF(FjtFjt-%+AXESLABELSG6$Q\"x6
\"Q!Fh[o-%%VIEWG6$;F(Fdt%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 20 "For example, taking " }{XPPEDIT 18 0 "x = 9/10;
" "6#/%\"xG*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 15 ", the sequence " }
{XPPEDIT 18 0 "phi[n](9/10) = (9/10)^n;" "6#/-&%$phiG6#%\"nG6#*&\"\"*
\"\"\"\"#5!\"\")*&F+F,F-F.F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n=1,2,3
,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 15 ", conve
rges to " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plot([
(9/10)^x,[seq([n,(9/10)^n],n=1..40)]],x=0..40,\n  symbol=circle,style=
[line,point],linestyle=2,color=[grey,blue]);" }}{PARA 13 "" 1 "" 
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\\&F`t7$$\"3\\mm;zM)>$GFbo$\"3/!QOr?d+1&F`t7$$\"3Z+++qfa<HFbo$\"3c(QR!
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$\"3;++++++hlF/7$$\"\"&F)$\"3f***********[!fF/7$$\"\"'F)$\"3_+++++T9`F
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[l6&F][lF(F($\"*++++\"F`[l-Fb[l6#%&POINTG-%*LINESTYLEG6#F]\\l-%+AXESLA
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ve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }
{XPPEDIT 18 0 "phi[n](1) = 1^n;" "6#/-&%$phiG6#%\"nG6#\"\"\")F*F(" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "n=1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"
#\"\"$%(~.~.~.~G" }{TEXT -1 28 ",  is the constant sequence " }
{XPPEDIT 18 0 "1, 1, 1, 1, ` . . . `" "6'\"\"\"F#F#F#%(~.~.~.~G" }
{TEXT -1 24 ", which converges to 1. " }}{PARA 0 "" 0 "" {TEXT -1 6 "S
ince " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT 
-1 21 " is continuous at 1, " }{XPPEDIT 18 0 "phi[n](x)->1" "6#f*6#-&%
$phiG6#%\"nG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F0F0F0" }{TEXT -1 
5 ", as " }{XPPEDIT 18 0 "x-> 1" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6
\"\"\"\"F*F*F*" }{TEXT -1 19 ", independently of " }{TEXT 284 1 "n" }
{TEXT -1 78 ". This means that we cannot obtain a maximum value for th
e difference between " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6
#%\"xG" }{TEXT -1 59 " and 0, and consequently the convergence cannot \+
be uniform." }}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }{XPPEDIT 
18 0 "Int(phi[n](x),x = 0 .. 1) = Int(x^n,x = 0 .. 1);" "6#/-%$IntG6$-
&%$phiG6#%\"nG6#%\"xG/F-;\"\"!\"\"\"-F%6$)F-F+/F-;F0F1" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 "1/(n+1)" "6#*&\"\"\"F$,&%\"nGF$F$F$!\"\"" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "n=1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$
%(~.~.~.~G" }{TEXT -1 18 ",  converges to 0." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the sequen
ce of functions " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"
\"$%(~.~.~.~G" }{TEXT -1 26 ", defined on the interval " }{XPPEDIT 18 
0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 5 " by: " }}{PARA 256 "" 0 
"" {TEXT -1 3 "   " }{XPPEDIT 18 0 "phi[n](x) = sum(matrix([[n], [k]])
*(1-abs(2*k/n-1))*((1+x)/2)^k*((1-x)/2)^(n-k),k = 0 .. n);" "6#/-&%$ph
iG6#%\"nG6#%\"xG-%$sumG6$**-%'matrixG6#7$7#F(7#%\"kG\"\"\",&F6F6-%$abs
G6#,&*(\"\"#F6F5F6F(!\"\"F6F6F>F>F6)*&,&F6F6F*F6F6F=F>F5F6)*&,&F6F6F*F
>F6F=F>,&F(F6F5F>F6/F5;\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1-``;" "6#/%!G,&\"\"\"F&F$!\"\"" }
{XPPEDIT 18 0 "Sum(matrix([[n], [k]])*abs(2*k/n-1)*((1+x)/2)^k*((1-x)/
2)^(n-k),k = 0 .. n);" "6#-%$SumG6$**-%'matrixG6#7$7#%\"nG7#%\"kG\"\"
\"-%$absG6#,&*(\"\"#F/F.F/F,!\"\"F/F/F6F/)*&,&F/F/%\"xGF/F/F5F6F.F/)*&
,&F/F/F:F6F/F5F6,&F,F/F.F6F//F.;\"\"!F," }{TEXT -1 2 ", " }}{PARA 0 "
" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "matrix([[n], [k]]);" "6#-%'
matrixG6#7$7#%\"nG7#%\"kG" }{TEXT -1 30 " is the binomial coefficient \+
 " }{XPPEDIT 18 0 "n!/(k!*(n-k)!)" "6#*&-%*factorialG6#%\"nG\"\"\"*&-F
%6#%\"kGF(-F%6#,&F'F(F,!\"\"F(F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG
6#%\"nG6#%\"xG" }{TEXT -1 8 " is the " }{TEXT 283 1 "n" }{TEXT -1 4 " \+
th " }{TEXT 261 20 "Bernstein polynomial" }{TEXT -1 17 " associated wi
th " }{XPPEDIT 18 0 "g(x) = 1-abs(x);" "6#/-%\"gG6#%\"xG,&\"\"\"F)-%$a
bsG6#F'!\"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]
" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 
"The sequence " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%
(~.~.~.~G" }{TEXT -1 12 ", converges " }{TEXT 261 9 "pointwise" }
{TEXT -1 20 " to the function g. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "alias(C=binomial):\nphi := \+
(n,x) -> 1-sum(C(n,k)*abs(2*k/n-1)*((1+x)/2)^k*((1-x)/2)^(n-k),k=0..n)
;\nplot([1-abs(x),phi(2,x),phi(4,x),phi(8,x),phi(16,x),phi(32,x),phi(6
4,x),\n  phi(128,x),phi(256,x),phi(512,x)],x=-1..1,color=[black,red,bl
ue,green,\n  magenta,coral,cyan,brown,COLOR(RGB,.4,0,.9),COLOR(RGB,.6,
.4,0)],\n  linestyle=[2,1$9]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$p
hiGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF),&\"\"\"F.-%$sumG6$**-%\"CG
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HF/7$Fay$\"3aPL$3Pls[#F/7$Ffy$\"3\"3+++I725#F/7$F[z$\"3'HLL$e)ywl\"F/F
_z7$Fez$\"394++vjM*Q)F27$Fjz$\"3!)3++D'zbM%F2F^[l-F[jp6&Fd[l$\"\"'F)F]
jpF*F^el-%+AXESLABELSG6$Q\"x6\"Q!Fjeq-%%VIEWG6$;F(F_[l%(DEFAULTG" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9
" "Curve 10" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 20 "For example, takin
g " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 24 ", we obtain \+
the sequence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[n](
0)=1-1/2^n" "6#/-&%\"fG6#%\"nG6#\"\"!,&\"\"\"F,*&F,F,)\"\"#F(!\"\"F0" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(matrix([[n],[k])*abs(1-2*k/n),k = \+
0 .. n)" "6#-%$SumG6$*&-%'matrixG6#7$7#%\"nG7#%\"kG\"\"\"-%$absG6#,&F/
F/*(\"\"#F/F.F/F,!\"\"F6F//F.;\"\"!F," }{TEXT -1 2 ". " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(phi(n,0));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&\"\"\"F$*&)\"\"#,$%\"nG!\"\"F$-%$sumG6$*&-%\"CG6
$F)%\"kGF$-%$absG6#*&,&*&F'F$F2F$F*F)F$F$F)F*F$/F2;\"\"!F)F$F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "This sequence appears to tend t
o 1, although the convergence is rather slow." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "phi := (n,x
) -> 1-sum(C(n,k)*abs(2*k/n-1)*((1+x)/2)^k*((1-x)/2)^(n-k),k=0..n):\np
lot([1,[[1,phi(1,0)],seq([5*n,phi(5*n,0)],n=1..20)]$2],0..100,\n  symb
ol=circle,style=[line$2,point],linestyle=[3,2],color=[cyan,grey,blue])
;\n" }}{PARA 13 "" 1 "" {GLPLOT2D 385 207 207 {PLOTDATA 2 "6(-%'CURVES
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LL$3dg6<*F<F*7$$\"3]nmmmxGp$*F<F*7$$\"3K,+D\"oK0e*F<F*7$$\"3Y,+v=5s#y*
F<F*7$$\"$+\"F)F*-%'COLOURG6&%$RGBGF($\"*++++\"!\")F\\u-%&STYLEG6#%%LI
NEG-%*LINESTYLEG6#\"\"$-F$6&777$F*F(7$$\"\"&F)$\"3+++++++]i!#=7$$\"#5F
)$\"3+++++D1RvF`v7$$\"#:F)$\"3+++]PMF0zF`v7$$\"#?F)$\"3vo/)*z%H!Q#)F`v
7$$\"#DF)$\"3!zeF?U(>)Q)F`v7$$\"#IF)$\"3=?j0>bNb&)F`v7$$\"#NF)$\"3x:oS
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F)$\"3_HySt#[s())F`v7$$\"#bF)$\"3,%\\Z5^J#>*)F`v7$$\"#gF)$\"3q/V\"*p#=
U(*)F`v7$$\"#lF)$\"3NJ._iC`1!*F`v7$$\"#qF)$\"3/BYfk_u\\!*F`v7$$\"#vF)$
\"3K()\\7whgv!*F`v7$$\"#!)F)$\"3CG4E77s5\"*F`v7$$\"#&)F)$\"34mfCAO-K\"
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\"3(G@Ghi2T?*F`v-Fit6&F[u$\")=THvF^uF`\\lF`\\lF_u-Fdu6#\"\"#-F$6&Fiu-F
it6&F[uF(F(F\\u-F`u6#%&POINTGFcu-%+AXESLABELSG6$Q!6\"F_]l-%'SYMBOLG6#%
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Consecutive pairs of
 terms of the sequence are equal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "phi(32,0);\nphi(33,0);\nphi(
44,0);\nphi(45,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"+`M%p%=\"+[O[
Z@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"+`M%p%=\"+[O[Z@" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6##\".(3&)3,O>\"._bDB!*>#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\".(3&)3,O>\"._bDB!*>#" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "a := n ->1-sum(evalf(binomial(n,k))*evalf(abs(1-2*k/n
)/2^n),k=0..n);\na(500);\na(1000);\na(2000);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&\"\"\"F--%$s
umG6$*&-%&evalfG6#-%\"CG6$9$%\"kGF--F36#*&-%$absG6#,&F-F-*(\"\"#F-F9F-
F8!\"\"FCF-)FBF8FCF-/F9;\"\"!F8FCF(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+WNNV'*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=)\\xu*!#5
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)))4;#)*!#5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "Now we investigate whether the sequence \+
" }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~
G" }{TEXT -1 12 ", converges " }{TEXT 261 9 "uniformly" }{TEXT -1 12 "
 to g where " }{XPPEDIT 18 0 "g(x) = 1-abs(x);" "6#/-%\"gG6#%\"xG,&\"
\"\"F)-%$absG6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 
"We can plot the difference between a function " }{XPPEDIT 18 0 "phi[n
](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
g(x) = 1-abs(x);" "6#/-%\"gG6#%\"xG,&\"\"\"F)-%$absG6#F'!\"\"" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "For example, the maximum diffe
rence between " }{XPPEDIT 18 0 "phi[100](x);" "6#-&%$phiG6#\"$+\"6#%\"
xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "1-abs(x);" "6#,&\"\"\"F$-%$abs
G6#%\"xG!\"\"" }{TEXT -1 15 " is about 0.08." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot((1-abs(x))-ph
i(100,x),x=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 415 116 116 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "It appears that the maximum dif
ference always occurs at " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }
{TEXT -1 59 ", and we saw above that this maximum difference tends to \+
0." }}{PARA 0 "" 0 "" {TEXT -1 32 "This suggests that the sequence " }
{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }
{TEXT -1 12 ", converges " }{TEXT 261 9 "uniformly" }{TEXT -1 12 " to \+
g where " }{XPPEDIT 18 0 "g(x) = 1-abs(x);" "6#/-%\"gG6#%\"xG,&\"\"\"F
)-%$absG6#F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 26 "We compute some values of " }{XPPEDIT 18 
0 "Int(phi[n](x),x = -1 .. 1);" "6#-%$IntG6$-&%$phiG6#%\"nG6#%\"xG/F,;
,$\"\"\"!\"\"F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "alias(C=binomial):\nphi := \+
(n,x) -> 1-sum(C(n,k)*abs(2*k/n-1)*((1+x)/2)^k*((1-x)/2)^(n-k),k=0..n)
:\na := [seq(int(phi(n,x),x=-1..1),n=1..16)];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"aG72\"\"!#\"\"#\"\"$F'#\"\"%\"\"&F*#\"\"'\"\"(F-#\"
\")\"\"*F0#\"#5\"#6F3#\"#7\"#8F6#\"#9\"#:F9#\"#;\"#<" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "This suggests that " }
{XPPEDIT 18 0 "Int(phi[n](x),x = -1 .. 1) = PIECEWISE([(n-1)/n, `if n \+
is odd`],[n/(n+1), `if n is even`]);" "6#/-%$IntG6$-&%$phiG6#%\"nG6#%
\"xG/F-;,$\"\"\"!\"\"F1-%*PIECEWISEG6$7$*&,&F+F1F1F2F1F+F2%,if~n~is~od
dG7$*&F+F1,&F+F1F1F1F2%-if~n~is~evenG" }{TEXT -1 23 ", so that the seq
uence " }{XPPEDIT 18 0 "Int(phi[n](x),x = -1 .. 1);" "6#-%$IntG6$-&%$p
hiG6#%\"nG6#%\"xG/F,;,$\"\"\"!\"\"F0" }{TEXT -1 2 ", " }{XPPEDIT 18 0 
"n = 1,2,3,` . . . `" "6&/%\"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 
11 ", tends to " }{XPPEDIT 18 0 "Int(1-abs(x),x = -1 .. 1)=1" "6#/-%$I
ntG6$,&\"\"\"F(-%$absG6#%\"xG!\"\"/F,;,$F(F-F(F(" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "Code for picture" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 36 
"Code for uniform convergence picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 669 "fn := x -> x^3/30-x/5:
\ngn := x -> 0.15*sin(2*x)+0.03*cos(7*x):\np1:= plot([fn(x)+0.6,fn(x)+
0.8,fn(x)+1,fn(x)+gn(x)+0.8],\n x=-2..3,color=[COLOR(RGB,.3,.3,.6)$3,r
ed],\n   linestyle=[2,1,2,1],tickmarks=[0,0]):\nt1 := plots[textplot](
[[3.38,1.3,`+`],[3.37,0.9,`-`]],\n      font=[HELVETICA,10],color=blac
k):\nt2 := plots[textplot]([[3.15,1.3,`f`],[3.6,1.31,`e`],[3.15,1.12,`
f`],\n    [3.15,0.9,`f`],[3.58,0.91,`e`]],font=[SYMBOL,12],color=black
):\nt3 := plots[textplot]([-2.11,1.00,`N`],font=[HELVETICA,10],color=r
ed):\nt4 := plots[textplot]([-2.25,1.05,`f`],font=[SYMBOL,12],color=re
d):\nplots[display]([p1,t1,t2,t3,t4],view=[-2.25..3.6,-0.1..1.5],\n   \+
title=`uniform convergence`);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }