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{TEXT 205 21 "Pointwise Convergence" }}{PARA 0 "" 0 "" {TEXT 228 18 "A sequence \{f _n(x)" }{TEXT 228 45 "\} of functions from a set S to R is said to " }{TEXT 216 18 "converge pointwise" }{TEXT 228 52 " to a function f i f for every x in S we have \{" }{TEXT 228 55 "\} converges to f(x ) as n tends to infinity. \n" }}{PARA 0 "" 0 "" {TEXT 217 19 "Bounded Convergence" }{TEXT 228 1 "\n" }{TEXT 228 9 " \n" } {TEXT 228 11 "A sequence " }{TEXT 228 9 "\{f_n(x)\} " }{TEXT 228 46 "o f functions from a set S to R is said to " }{TEXT 218 18 "converge boundedly" }{TEXT 228 22 " to a function f if \{" }{TEXT 228 82 "\} \+ converges pointwise to f and it is possible to find a number K su ch that |" }{TEXT 228 46 "| <= K for every x in S and every n. \+ \n" }}{PARA 0 "" 0 "" {TEXT 219 19 "Uniform Convergence" }{TEXT 228 1 "\n" }{TEXT 228 9 " \n" }{TEXT 228 11 "A sequence " }{TEXT 228 8 "\{f_n(x)\}" }{TEXT 228 47 " of functions from a set S to R \+ is said to " }{TEXT 220 18 "converge uniformly" }{TEXT 228 28 " to a f unction f if sup |" }{TEXT 228 1 " " }{TEXT 221 4 "f(x)" }{TEXT 228 49 " | tends to 0 as n tends to infinity. \n" }}{PARA 0 "" 0 "" {TEXT 222 10 "Exercise 1" }{TEXT 228 46 ": Demonstrate that th e sequence of functions " }{TEXT 228 58 "is pointwise convergent to 0 but not uniformly convergent." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f :=(x,n)-> (2*n*x)/(1+n*x^2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(x,n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{f( x,5),f(x,10)\},x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 39 "We ca lculate the maximum of each f(x,n)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "solve(diff(f(x,n),x)=0,x);#We take the positive x;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "subs(x=sqrt(3)/(3*sqrt(n) ),f(x,n));#Note that this shows that the peaks of f(x,n) goes to infin ity. Therefore, it is easy to see that the sequence f(x,n) does not co verge uniformly." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(\{ f(x,10^9),f(x,10^10)\},x=0..10^(-8));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P:=(x,k)-> (10^k)*x*exp((-10^k)*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{P(x,12),P(x,11)\},x=0..10^(-10),y= 0..0.4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for k from 9 by 1 to 16 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "P||k:= plot (\{P(x,16),P(x,k) \}, x=0.." }{MPLTEXT 1 0 8 "10^(-10)" }{MPLTEXT 1 0 25 ", y=0..0.40,th ickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plots[display]([seq(P||k, k=9..16)], inse quence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 223 10 "Exercise 2" } {TEXT 224 1 ":" }{TEXT 228 58 " Demonstrate by animation that the sequ ence of functions " }{TEXT 228 25 "converges uniformly to 0." }} {PARA 0 "" 0 "" {TEXT 228 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=proc(x,k) k*x*exp((-k^2)*x) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for k from 1 by 1 to 20 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "f||k:= plot (\{f(x,20),f(x,k)\}, x=0..1, y=0..0.6,thickness=2):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plots[display]([seq(f||k, k=1..20)], insequence=true) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 10 "Exercise 3" }{TEXT 226 1 ": " }{TEXT 228 56 " Demonstrate by animation that the sequence of functi ons" }{TEXT 228 39 " converges pointwise but not boundedly." }}{PARA 0 "" 0 "" {TEXT 228 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f :=proc(x,k) k^2*x*exp(-k*x) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fo r k from 1 by 1 to 30 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "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 54 "plots[display]([seq(f||k, k=1..30)], insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 227 11 "Exercise 4:" }{TEXT 228 56 " Uniform convergence on functions \+ and differentiability:" }}{PARA 0 "" 0 "" {TEXT 228 257 "The following example shows that the uniform convergence of a sequence f(x,n) is no t enough to guarantee the limit of the derivative of f(x,n) converges. We note that following f(x,k) converges uniformly to 0 in [0,1] but l imit of f'(x,n) is not equal to 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:=(x,k)-> sin(k *x)/(sqrt(k)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(\{f(x,10),f(x,15),f(x,20)\},x=0..1,y=-1..1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{f(x,20),f(x ,25)\},x=0..1,y=-1..1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "plot(\{f(x,100),f(x,150),f(x,200)\},x=0..1,y=-1..1) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 205 60 "Can you conjecture if f(x,k ) converges pointwisely in [0,1]?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "for k from 150 by 1 to 200 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f||k:= plot (\{f(x,200),f(x,k)\}, x=0..1,y=-0.1..0.1, thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plots[display]([seq(f||k, k=150..20 0)], insequence=true);" }}}{EXCHG {PARA 0 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movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.3333333em\"), Typesetting:-mi(\"g\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mfenced(Typesetting:-mrow( Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), T ypesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", sep arator = \"true\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Typesetting:-mn(\"20\", mathvar iant = \"normal\")), mathvariant = \"normal\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \"true\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.3333333em\"), Typesetting:-mi(\"g\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-m i(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo (\",\", mathvariant = \"normal\", fence = \"false\", separator = \"tru e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.3333333em\"), Typesetting:-mn(\"100\", mathvariant = \"norm al\")), mathvariant = \"normal\")), mathvariant = \"normal\", open = \+ \"\{\", close = \"\}\"), Typesetting:-mo(\",\", mathvariant = \"normal \", fence = \"false\", separator = \"true\", stretchy = \"false\", sym metric = \"false\", largeop = \"false\", movablelimits = \"false\", ac cent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Types etting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"0\", mathva riant = \"normal\"), Typesetting:-mo(\"..\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.2222222em\", rspace = \"0.0em\"), Typeset ting:-mn(\"1\", mathvariant = \"normal\")), mathvariant = \"normal\"), Typesetting:-mo(\";\", mathvariant = \"normal\", fence = \"false\", s eparator = \"true\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.2777778em\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q%plotF'/%'italicGQ%trueF'/%,m athvariantGQ'italicF'-I(mfencedGF$6$-F#6)-F66&-F#6*-F,6%Q\"gF'F/F2-F66 $-F#6%-F,6%Q\"xF'F/F2-I#moGF$6-Q\",F'/F3Q'normalF'/%&fenceGQ&falseF'/% *separatorGF1/%)stretchyGFP/%*symmetricGFP/%(largeopGFP/%.movablelimit 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