{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 57 "Section \+ 2.1: Convergence of Infinite Sequences and Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 30 "Maple Packages f or Section 2.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 271 "In this section, we address the genera l notion of convergence in a function space. There is a larger variety of definitions of convergence than we shall discuss here. Rather, we \+ contrast three types of convergence in C([0, 1]): normed, pointwise, a nd uniform convergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " We suppose that we have a sequence of functions " }{XPPEDIT 18 0 "f[1](x);" "6#-&%\"fG6#\"\"\"6#%\"xG" }{TEXT -1 3 " \+ , " }{XPPEDIT 18 0 "f[2](x);" "6#-&%\"fG6#\"\"#6#%\"xG" }{TEXT -1 3 " \+ , " }{XPPEDIT 18 0 "f[3](x);" "6#-&%\"fG6#\"\"$6#%\"xG" }{TEXT -1 27 " , ... and another function " }{TEXT 268 1 "g" }{TEXT -1 1 "(" }{TEXT 279 1 "x" }{TEXT -1 19 "). We say that the " }{XPPEDIT 18 0 "f;" "6#% \"fG" }{TEXT -1 15 " 's converge to" }{TEXT 269 2 " g" }{TEXT -1 18 " \+ on the interval [" }{TEXT 273 1 "a" }{TEXT -1 2 ", " }{TEXT 272 1 "b" }{TEXT -1 17 "] in the sense of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "1. " }{TEXT 256 16 "norm convergence" } {TEXT -1 6 ", if " }{XPPEDIT 18 0 "int(abs(f[n](t)-g(t))^2,t = a .. b );" "6#-%$intG6$*$-%$absG6#,&-&%\"fG6#%\"nG6#%\"tG\"\"\"-%\"gG6#F1!\" \"\"\"#/F1;%\"aG%\"bG" }{TEXT -1 17 " -> 0 as n -> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " }{TEXT 257 11 "pointw ise, " }{TEXT -1 20 " if for each x in [" }{TEXT 275 1 "a" }{TEXT -1 2 ", " }{TEXT 274 1 "b" }{TEXT -1 6 "], " }{XPPEDIT 18 0 "f[n](x); " "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 19 " -> g(x) as n -> " } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3. " }{TEXT 258 11 "uniformly, " }{TEXT -1 30 " if the maximum for all x in [" }{TEXT 277 1 "a" }{TEXT -1 2 ", " }{TEXT 276 1 "b" }{TEXT -1 5 "] of " } {XPPEDIT 18 0 "abs(f[n](x)-g(x));" "6#-%$absG6#,&-&%\"fG6#%\"nG6#%\"xG \"\"\"-%\"gG6#F-!\"\"" }{TEXT -1 24 " goes to zero as n -> " } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 210 "As it might be ex pected these definitions of convergence are not all equivalent. Unifor m convergence is the strongest condition, in the sense that if a seque nce converges uniformly, then it will converge on C([" }{TEXT 280 1 "a " }{TEXT -1 2 ", " }{TEXT 278 1 "b" }{TEXT -1 327 "]) in the norm and \+ pointwise, too. When making a model, this strong uniform convergence i s to be desired. If that type of convergence does not hold, then one s hould get the strongest that is available. From what has come before, \+ we know that if we begin with an orthogonal sequence and generate a Fo urier Series for a function " }{TEXT 270 1 "f" }{TEXT -1 140 ", then t he Fourier Series will converge in norm.We will try to understand the \+ other two types of convergence in this and following sections." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "In order to give an understanding of the definitions of convergence, we compar e and contrast them for several sequences of functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }{TEXT 261 49 "U niform Convergence implies pointwise convergence" }{TEXT -1 34 ". To s ee this, note only that if " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 30 " " } {XPPEDIT 18 0 "MAX[x]*abs(f[n](x)-g(x));" "6#*&&%$MAXG6#%\"xG\"\"\"-%$ absG6#,&-&%\"fG6#%\"nG6#F'F(-%\"gG6#F'!\"\"F(" }{TEXT -1 9 " -> 0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "then \+ for each x, " }{XPPEDIT 18 0 "f[n](x)-g(x);" "6#,&-&%\"fG6#%\"nG6 #%\"xG\"\"\"-%\"gG6#F*!\"\"" }{TEXT -1 9 " -> 0." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " }{TEXT 262 47 "Uniform Convergence implies normed con vergence." }{TEXT -1 24 " To see this, note that " }}{PARA 0 "" 0 "" {TEXT -1 27 " " }}{PARA 0 "" 0 "" {TEXT -1 26 " " }{XPPEDIT 18 0 "abs(int((f[n](x)-g(x)) ^2,x = 0 .. 1)) <= MAX[x]*abs(f[n](x)-g(x))^2;" "6#1-%$absG6#-%$intG6$ *$,&-&%\"fG6#%\"nG6#%\"xG\"\"\"-%\"gG6#F2!\"\"\"\"#/F2;\"\"!F3*&&%$MAX G6#F2F3*$-F%6#,&-&F.6#F06#F2F3-F56#F2F7F8F3" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3. " }{TEXT 263 57 "Pointwise convergence does not im ply uniform convergence." }{TEXT -1 176 " To see this, note that the f ollowing sequence of functions converges to the zero function g(x)=0 p ointwise, but the maximum of each function differs from the zero funct ion by " }}{PARA 0 "" 0 "" {TEXT -1 64 " \+ 1/e = 0.367689... ." }}{PARA 0 "" 0 "" {TEXT -1 78 "How shall the sequence of functions be displayed? One way is to plot them all." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([seq(n*x*exp(- n*x),n=1..10)],x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 235 "An alternate method of displaying the sequence is to show the terms of the sequenc e one term at a time. After executing the following statement, animate the plot by touching with the mouse the icon for \"move to the next f rame\" command." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display([ seq(plot(n*x*exp(-n*x),x=0..1),n=1..20)],insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 160 "F inally, I try to convince you that the sequence converges to zero poin twise by choosing one point and letting you observe the passage to nea rly zero. I choose " }{TEXT 271 1 "x" }{TEXT -1 7 " = 0.8." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for n from 1 to 15 do\n n*0.8*exp (-n*0.8);\nend do;\nn:='n':" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "4. " }{TEXT 264 55 "Pointwise convergence does not imply normed convergence" }{TEXT -1 20 ". To see this, take " }{XPPEDIT 18 0 "f[n];" "6#&%\"fG6#%\"nG" }{TEXT -1 24 " to be defined this way" }}{PARA 0 "" 0 "" {TEXT -1 34 " " }{XPPEDIT 18 0 "f[n](x);" "6#-&% \"fG6#%\"nG6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "n^2*x*exp(-n*x); " "6#*(%\"nG\"\"#%\"xG\"\"\"-%$expG6#,$*&F$F'F&F'!\"\"F'" }{TEXT -1 6 " , if " }{XPPEDIT 18 0 "x <> 0;" "6#0%\"xG\"\"!" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 160 "We illustrate with graphs that the seque nce converges to zero pointwise, but the integral of the difference sq uared goes to infinity. First, we draw the graphs." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 69 "display([seq(plot(n^2*x*exp(-n*x),x=0..1),n= 1..10)],insequence=true);" }}}{PARA 0 "" 0 "" {TEXT -1 129 "To see, fo r sure, that the sequence converges pointwise, take a positive x and e valuate what happens as n increases without bound" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "assume(x>0);\nlimit(n^2*x*exp(-n*x),n=infinit y);\nx:='x':" }}}{PARA 0 "" 0 "" {TEXT -1 107 "Now, to see that the se quence does not converge in norm, we evaluate the integral and let n g o to infinity." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Int((n^2*x *exp(-n*x))^2,x=0..1)=int((n^2*x*exp(-n*x))^2,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(%,n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 3 "5. " }{TEXT 265 53 "Norm convergence does not imply pointwise convergence" }{TEXT -1 130 ". The following is ni ne terms of an infinite sequence. The sequence does not converge point wise. To see this, after executing the " }{TEXT 259 7 "display" } {TEXT -1 320 " command, touch the graph with the mouse, see a new tool bar above; the tool bar looks like a CD player control. Push the ->| \+ symbol with the mouse. The graphs will progress through in order. You \+ should see that you do not have pointwise convergence. To see that the norm converges to zero, compute the norm in the next " }{TEXT 260 7 " do loop" }{TEXT -1 36 ". See how the computations might go." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "f[1]:=x->(1+ signum(1/2-x))/2:\nf[2]:=x->(1+signum(x-1/2))/2:\nf[3]:=x->(1+signum(1 /3-x))/2:\nf[4]:=x->(1+signum(2/3-x))/2-(1+signum(1/3-x))/2:\nf[5]:=x- >(1+signum(1-x))/2-(1+signum(2/3-x))/2:\nf[6]:=x->(1+signum(1/4-x))/2- (1+signum(0-x))/2:\nf[7]:=x->(1+signum(1/2-x))/2-(1+signum(1/4-x))/2: \nf[8]:=x->(1+signum(3/4-x))/2-(1+signum(1/2-x))/2:\nf[9]:=x->(1+signu m(1-x))/2-(1+signum(3/4-x))/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for n from 1 to 9 do\n p[n]:=plot([f[n](x)],x=0..1):\nend do :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "display([seq(p[n],n=1. .9)],insequence=true);" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for n from 1 to 9 do\nint(f[n](x)^2 ,x=0..1);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "We have introduced these notio ns of convergence in the context of sequences. A series is a sequence. After all, one has only to identify " }}{PARA 0 "" 0 "" {TEXT -1 26 " " }{XPPEDIT 18 0 "f[n](x);" "6#-&%\"fG6#%\"n G6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sum(a[p]*sin(p*Pi*x),p = 1 .. n);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"-%$sinG6#*(F*F+%#PiGF+%\"xGF +F+/F*;F+%\"nG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "to conv ert these statements of convergence of sequences to statements about c onvergence of series. In the next Section, we will discuss how series \+ converge with this language." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: herod@ma th.gatech.edu or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "UR L: http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 36 "Copyright \251 2003 by James V. Her od" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }