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{SECT 0 {PARA 0 "" 0 "" {TEXT 256 36 "Convergence of Series, With Exam
ples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 330 "
I would like to say some things about convergence of a series of funct
ions. Three examples will be given which will illustrate the ideas. On
e of the series does not converge in norm, one converges pointwise but
not uniformly, and one converges uniformly. It may be that you should
get out your old calculus book to review series. " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 277 "In the Lecture 8, I disc
ussed general notions of convergence for a sequence of functions and, \+
in Lecture 9, I considered the special case that the sequence of funct
ions came from a Fourier Series. I expand on this special case. I cons
ider a Fourier Series expressed in the form" }}{PARA 0 "" 0 "" {TEXT
-1 31 "( SERIES MODEL ) " }{XPPEDIT 18 0 "sum(a[p]*sin(p
*x),p = 1 .. infinity);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"-%$sinG6#*&F
*F+%\"xGF+F+/F*;F+%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 241 "Here are four results yo
u should know. There are two results which give a criteria for converg
ence based on information about the coefficients and two results which
give criteria for convergence based on information about the limit fu
nction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257
9 "Result 1." }{TEXT -1 5 " The " }{TEXT 265 12 "series model" }{TEXT
-1 55 " converges in norm if and only if the number sequence \{" }
{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT
18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[3];
" "6#&%\"aG6#\"\"$" }{TEXT -1 30 " , ... \} has the property that" }}
{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "sum(abs(a[p])
^2,p = 1 .. infinity);" "6#-%$sumG6$*$-%$absG6#&%\"aG6#%\"pG\"\"#/F-;
\"\"\"%)infinityG" }{TEXT -1 4 " < " }{XPPEDIT 18 0 "infinity;" "6#%)
infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT 258 9 "Result 2." }{TEXT -1 26 " If the number sequence
\{" }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT
18 0 "a[3];" "6#&%\"aG6#\"\"$" }{TEXT -1 30 " , ... \} has the propert
y that" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "sum
(abs(a[p]),p = 1 .. infinity);" "6#-%$sumG6$-%$absG6#&%\"aG6#%\"pG/F,;
\"\"\"%)infinityG" }{TEXT -1 4 " < " }{XPPEDIT 18 0 "infinity;" "6#%)
infinityG" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 9 "then the " }
{TEXT 268 12 "series model" }{TEXT -1 21 " converges uniformly." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 9 "Result 3.
" }{TEXT -1 71 " If the limit function is sectionally smooth and perio
dic with period 2" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 11 ", then
the " }{TEXT 266 12 "series model" }{TEXT -1 41 " converges at every \+
x. The limit will be " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\""
}{TEXT -1 19 " [ f(x+) + f(x-) ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 260 9 "Result 4." }{TEXT -1 52 " If the periodic
extension of the limit function is " }{TEXT 269 3 "not" }{TEXT -1 17
" continuous, the " }{TEXT 267 12 "series model" }{TEXT -1 29 " does n
ot converge uniformly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 110 "Be sure you understand the difference between a Fou
rier series converging pointwise and converging uniformly. " }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "Indicat
ions of Proofs for these Results." }}{PARA 0 "" 0 "" {TEXT 261 9 "Resu
lt 1:" }{TEXT -1 89 " This result can be established by expanding the \+
square of the norm for the model series." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 41 "int(sum(a[p]*sin(p*x),p=1..5)^2,x=0..Pi);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Thu
s, " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "abs(su
m(a[p]*sin(p*x),p))^2;" "6#*$-%$absG6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"-%
$sinG6#*&F.F/%\"xGF/F/F.\"\"#" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "Pi/
2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sum(a
bs(a[p])^2,p);" "6#-%$sumG6$*$-%$absG6#&%\"aG6#%\"pG\"\"#F-" }{TEXT
-1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 "Result 2" }{TEXT -1
112 ": We have only to see that sums from m to n get small independent
of x. Because sin(x) is no larger than 1, then" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18
0 "abs(sum(a[p]*sin(p*x),p = m .. n)) <= sum(abs(a[p]),p = m .. n);" "
6#1-%$absG6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"-%$sinG6#*&F.F/%\"xGF/F//F.;
%\"mG%\"nG-F(6$-F%6#&F,6#F./F.;F7F8" }{TEXT -1 3 " ." }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Consequently if the s
um on the right gets small the the sum on the left gets small independ
ent of x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
263 9 "Result 3." }{TEXT -1 145 " A proof of this result in included i
n the book by Powers, Boundary Value Problems, Third Edition, page 79
- 83 or Fourth Edition, page 94 - 99." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 264 8 "Result 4" }{TEXT -1 252 ". This resul
t follows from the notion that sequences of continuous functions which
converge uniformly converge to a continuous function. Thus, if the li
mit of a sequence of continuous functions is not continuous, then the \+
convergence cannot be uniform." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33
"I now present three examples. In " }{TEXT 270 17 "the first example"
}{TEXT -1 73 ", we do not have convergence in norm. There is no functi
on f(x) such that" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }
{XPPEDIT 18 0 "int(abs(f(x))^2,x = 0 .. Pi) < infinity;" "6#2-%$intG6$
*$-%$absG6#-%\"fG6#%\"xG\"\"#/F.;\"\"!%#PiG%)infinityG" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 52 "to which the series converges. This i
s true because " }{XPPEDIT 18 0 "a[p] = 1;" "6#/&%\"aG6#%\"pG\"\"\"" }
{TEXT -1 83 " and the infinite sum of 1's is not finite. (See Result 1
.) The first example is " }{XPPEDIT 18 0 "sum(sin(n*x),n = 1 .. infi
nity);" "6#-%$sumG6$-%$sinG6#*&%\"nG\"\"\"%\"xGF+/F*;F+%)infinityG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 89 "f1:=x->sum(sin(n*x),n=1..15):\ng1:=x->sum(sin(n*
x),n=1..25):\nh1:=x->sum(sin(n*x),n=1..35):" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 70 "plot([f1(x),g1(x),h1(x)],x=-Pi..Pi,y=-2..2,\n col
or=[red,blue,green]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 22 "The second example is " }{XPPEDIT 18 0 "sum(sin(n*x)
/n,n = 1 .. infinity);" "6#-%$sumG6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF,F,F
+!\"\"/F+;F,%)infinityG" }{TEXT -1 145 ". I try to convince you and me
that the convergence is not uniform by seeing that the series converg
es in norm, but not to a continuous function." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 95 "f2:=x->sum(sin(n*x)/n,n=1..15):\ng2:=x->sum(sin(
n*x)/n,n=1..25):\nh2:=x->sum(sin(n*x)/n,n=1..35):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 71 "plot(\{f2(x),g2(x),h2(x)\},x=-Pi..Pi,y=-2..2
,\n color=[red,blue,green]);" }}}{PARA 0 "" 0 "" {TEXT -1 83 "I know
this example converges in norm because of the following sum. (See Res
ult 1.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(1/n^2,n=1..in
finity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The third example is
" }{XPPEDIT 18 0 "sum(sin(n*x)/(n^2),n = 1 .. infinity);" "6#-%$sumG6
$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF,F,*$F+\"\"#!\"\"/F+;F,%)infinityG" }
{TEXT -1 82 ". I try to convince you and me that the series converges \+
to a continuous function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
101 "f3:=x->sum(sin(n*x)/n^2,n=1..15):\ng3:=x->sum(sin(n*x)/n^2,n=1..2
5):\nh3:=x->sum(sin(n*x)/n^2,n=1..35):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 36 "plot(\{f3(x),g3(x),h3(x)\},x=-Pi..Pi);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 140 "T
his suggests work. We PROVE that the last series converges uniformly a
s our intuition suggests. See Result 2. To use this result, identify \+
" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 85 " and recall \+
that sums of such terms converge. In fact, Maple knows the infinite su
m." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(1/n^2,n=1..infinit
y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 166 "This proves that the last Fourier series converges unifo
rmly. That is, because the Fourier Coefficients are summable, then the
Fourier Series is uniformly convergent." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Now, who knows what the last two s
eries converges to?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 157 "I added these lecture not
es Section 9b for I think some specific examples improves understandin
g. If you had more time, you could have made them up yourself." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 315 "There ar
e folks not taking this class who study these notes for their pleasure
... really. The notes are free to the world but, as a challenge and t
o let me know about your pleasure, why don't you folks out there in cy
ber space make some interesting examples to contrast these ideas and t
ell me about your examples." }}}{MARK "0 0" 36 }{VIEWOPTS 1 1 0 1 1
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }