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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 263 66 "Maple Illustrations of S
elected Topics from Undergraduate Analysis" }}{PARA 19 "" 0 "" {TEXT 
-1 13 "J. A. Ziegler" }}{PARA 256 "" 0 "" {TEXT -1 37 "Southern Polyte
chnic State University" }}{PARA 256 "" 0 "" {TEXT 264 44 "http://www2.
SPSU.edu/math/ziegler/index.html" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 506 "
An undergraduate analysis course has several important goals, in addit
ion to a reasonable mastery of its contents. One of the most important
 too often is almost omitted because \"there just isn't time.\" This i
s a serious attempt to help students see \"how they can do it, too,\" \+
by explicitly discussing, in specific instances, the interplay among i
ntuition, the exploration of examples and special cases, and the const
ruction of a proof.  In the three non-trivial instances below, Maple c
an be very helpful. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 35 "  The Riemann Rearrangement Theorem" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 634 "Most students are greatly surpris
ed to hear that a conditionally convergent series can be rearranged to
 converge to any desired sum, or even to diverge.  Before the reasons \+
are explained to them, these facts can be made plausible by using Mapl
e to illustrate them.  The procedure below follows that of the proof a
nd so can be used to reconstruct it.  The relevant subsidiary question
s then arise naturally and the ways by which they may be answered are \+
better appreciated.  In this way, the complementary roles of intuition
 and proof are nicely illustrated and the student has the experience o
f participating in mathematical research." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 68 "To illustrate the Maple procedure, the alternating harmon
ic series  " }{XPPEDIT 18 0 "sum((-1)^(n+1)/n,n = 1 .. infinity)" "6#-
%$sumG6$*&),$\"\"\"!\"\",&%\"nGF)F)F)F)F,F*/F,;F)%)infinityG" }{TEXT 
-1 89 "   has been used, but the procedure can be used with any condit
ionally convergent series." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 294 "Th
e Maple procedure we now define imitates the proof of Riemann's theore
m.  Given a desired value, S, and the number, N,  of terms to use,  it
  uses the approximation procedure described in Riemann's proof to par
tially create a suitable rearrangement of the original series.  The su
m of this,  " }{XPPEDIT 18 0 "S[N];" "6#&%\"SG6#%\"NG" }{TEXT -1 547 "
,  is as \"close\" to S as it is possible to get using only N terms an
d is the Nth partial sum of the completely rearranged series.  The Map
le procedure also determines an \"error\", which is the magnitude of t
he largest still unused term from the original series.  The output is \+
both numerical and graphical.  The numerical output shows the terms us
ed, in order, and reports the \"error\" and the numbers of positive an
d negative terms used.  The graphical output shows S, an \"error\" ban
d about S, and the partial sums of the rearranged series through " }
{XPPEDIT 18 0 "S[N];" "6#&%\"SG6#%\"NG" }{TEXT -1 97 " , connected by \+
line segments to create a \"curve\" showing the progress of the proced
ure toward S." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:wi
th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2410 "rearr:=proc
(x,N)\n          local b,S,apprx,i,j,k,r,m,m1,e,p1,p2,pS,terms,psums,S
n:\n\nS:=evalf(x):\n    apprx:=0:b:='b':\n    m:=1:m1:=1:\n    for i f
rom 1 to N do\n\nif apprx<=S then b[i]:=1/(2*m-1);# If the current val
ue of the approximation does not exceed the                           \+
    value of the goal, let the next term in the rearranged series be t
he next unused positive term.                                         \+
                                                                      \+
                                                                      \+
              \napprx:=sum(b[j],j=1..i);#        Add this to the previ
ous partial sum.\n \nm:=m+1:#                         Increase the cou
nt of positive terms used by one.\n\nelse b[i]:=-1/(2*m1);#         Ot
herwise, let the next term in the rearranged series be the next unused
 negative term.\n\napprx:=sum(b[j],j=1..i);#        Add this to the pr
evious partial sum.\n\nm1:=m1+1:#                       Increse the co
unt of negative terms used by one.\nfi:\nod:\n\nterms:=[seq(b[k],k=1..
N)]:#      Collect the first N terms of the sequence of terms of the  \+
                               rearranged series.\n\nSn:=n->sum(b[s],s
=1..n):#        Define the nth partial sum of the rearranged series.\n
\npsums:=[seq([r,Sn(r)],r=1..N)]:# Form the sequence of partial sums o
f the rearranged series.\n\nif m<=m1 then e:=1/(2*m-1);#        If few
er positive terms than negative ones have been used,let the \"error\" \+
be the next positive term.\n\nelse e:=1/(2*m1);#             Otherwise
, let this be the magnitude of the next negative term.\nfi:\n\nlprint(
terms);#                  Print in order the first N terms of the rear
ranged series.\nlprint():\nlprint(`number of positive terms used =`,m-
1):\nlprint(`number of negative terms used =`,m1-1):\nlprint(`the Nth \+
partial sum is about`,evalf(Sn(N),3)):\nlprint(`maximum error =`,e):\n
lprint():\n\np1:=plot(psums,0..N,S-0.2..S+0.2,style=line,color=blue):#
 Draw a \"curve\" to represent the                                    \+
                       progress of the rearranged series\n\np2:=plot([
S-e,S+e],0..N,S-0.2..S+0.2,color=black):#       Draw lines to show the
 \"error band\"                                                       \+
                                                              after N \+
steps.\n\npS:=plot(S,0..N,S-0.2..S+0.2,style=line,thickness=2,color=re
d):# Draw a line to show the goal.\n\ndisplay(p1,p2,pS);\nend:        \+
  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "To check our procedure, let
 S = ln(2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rearr(ln(2),
50);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "As an example, let us ha
ve Maple rearrange our series to converge to S = 1, using 100 terms of
 the rearranged series. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"rearr(1,100);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 "For any partic
ular choice of S, what pattern of positive and negative terms may one \+
expect?  One of my current students tried S = 0, by hand, and found, f
or a few terms, a pattern of 1 positive term followed by 4 negative on
es.  Let us try it for 200 terms.  A careful inspection shows that he \+
is correct, so far." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "rear
r(0,200);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 138 "Note: A somewhat similar, and simplier, \+
treatment may be found in M. Holmes, et. al., Exploring Calculus with \+
Maple, Addison-Wesley, 1993." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 37 "  Uniform and Non-Uniform Convergence" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 528 "Students in an undergraduate anal
ysis course often find uniform convergence both unexpected and somewha
t difficult.  Of course, uniform convergence did not occur to Cauchy, \+
either, and the students may be encouraged to take comfort from this. \+
 However, if Cauchy had had Maple with which to try out ideas, one rat
her imagines uniform convergence would not have been overlooked.  Mapl
e animations provide a particularly clear introduction.  Without Maple
, the best treatment known to me, with plenty of graphs, remains Coura
nt's " }{TEXT 257 36 "Differential and Integral Calculus. " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 187 "The Maple procedure given can be used fo
r many questions of uniform convergence.  With its aid, one may quickl
y decide whether to try to prove that the convergence is or is not uni
form.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plo
ts):with(plottools):" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 
256 0 "" }{TEXT -1 38 "Example 1.  The sequence of functions " }
{XPPEDIT 18 0 "f[n](x) = x^n;" "6#/-&%\"fG6#%\"nG6#%\"xG)F*F(" }{TEXT 
-1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "We begin by defining o
ur functions.  For technical reasons, we will omit the subscripts.  " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=(n,x)->x^n;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p:=n->plot(f(n,x),x=0..1,col
or=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "epsilon:=0.1;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:=0.9:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 81 "In addition to the graph of y = f(n,x), w
e include in our picture a line to show " }{XPPEDIT 18 0 "epsilon;" "6
#%(epsilonG" }{TEXT -1 50 " and one to show b.  We also include the va
lue of " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 28 " and t
he current value of n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "E
:=plot(epsilon,x=0..1,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "X:=line([b,0],[b,1],color=green):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 128 "T:=n->textplot(\{[.5,.8,cat(\"n = \", n)],
[.5,.9,cat(`epsilon = `,convert(epsilon,string))]\},align=ABOVE,font=[
HELVETICA,BOLD,12]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pic
:=seq(display(T(n),p(n),X,E),n=1..50):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "display(pic,insequence=true,view=[0..1.5,0..1],thickn
ess=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 147 "We now click on the picture and run the \+
animation one frame at a time.  We see that for n > 22, the graph of y
 = f(n,x) lies between y = 0 and y = " }{XPPEDIT 18 0 "epsilon;" "6#%(
epsilonG" }{TEXT -1 10 " on [0,b]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 31 "By increasing b and decreasing " }{XPPEDIT 18 0 "epsilon;" "6#%
(epsilonG" }{TEXT -1 187 ", we may actually see that it is plausible t
hat our sequence of functions converges uniformly to 0 on [0, b] so lo
ng as b < 1.  We also see that this is not going to be the case on [0,
1]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "We see something else wor
th remembering.  On [0,1], the limit function of our sequence of conti
nuous functions is discontinuous.  Thus it has a property which is not
 shared by any of the terms of the sequence." }}}{EXCHG {PARA 4 "" 0 "
" {TEXT -1 40 "Example 2.  The sequence of functions f(" }{XPPEDIT 18 
0 "alpha;" "6#%&alphaG" }{TEXT -1 8 ",n,x) = " }{XPPEDIT 18 0 "x*n^alp
ha*exp(-n*x);" "6#*(%\"xG\"\"\")%\"nG%&alphaGF%-%$expG6#,$*&F'F%F$F%!
\"\"F%" }{TEXT -1 17 " for fixed, real " }{XPPEDIT 18 0 "alpha;" "6#%&
alphaG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Here th
e nature of the convergence on [0,b], for 0 < b, depends on " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 87 ", although this is n
ot exactly obvious to the beginner.  The convergence is uniform if " }
{XPPEDIT 18 0 "alpha < 1;" "6#2%&alphaG\"\"\"" }{TEXT -1 20 " and non-
uniform if " }{XPPEDIT 18 0 "1 <= alpha;" "6#1\"\"\"%&alphaG" }{TEXT 
-1 40 ". We illustrate this with two examples: " }{XPPEDIT 18 0 "alpha
;" "6#%&alphaG" }{TEXT -1 11 " = 1/2 and " }{XPPEDIT 18 0 "alpha" "6#%
&alphaG" }{TEXT -1 30 " = 3/2.  We will choose b = 2." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=(alpha,n,x)->x*n^alpha*exp(-n*x)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "p:=n->plot(f(alpha,n,x
),x=0..2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ep
silon:=0.1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b:=2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "E:=plot(epsilon,x=0..2,color
=red):" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "Case 1: " }{XPPEDIT 18 
0 "alpha < 1;" "6#2%&alphaG\"\"\"" }{TEXT -1 3 "   " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "alpha:=1/2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 127 "T:=n->textplot(\{[1,.2,cat(\"n = \", n)],[1,.24,cat(
`epsilon = `,convert(epsilon,string))]\},align=ABOVE,font=[HELVETICA,B
OLD,12]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "pic:=seq(displ
ay(T(n),p(n),E),n=1..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "display(pic,insequence=true,thickness=2);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" 
{TEXT -1 9 "Case 2:  " }{XPPEDIT 18 0 "1 <= alpha;" "6#1\"\"\"%&alphaG
" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "alpha:=3
/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "p:=n->plot(f(alpha,n
,x),x=0..2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 
"T2:=n->textplot(\{[1,1,cat(\"n = \", n)],[1,1.3,cat(`epsilon = `,conv
ert(epsilon,string))]\},align=ABOVE,font=[HELVETICA,BOLD,12]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pic2:=seq(display(T2(n),p(n)
,E),n=1..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(pi
c2,insequence=true,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 72 "  Weierstra
ss' Everywhere Continuous and Nowhere Differentiable Function" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "Because we only have a little tim
e today, I will not mention the interesting history of Weierstrass' (1
872) everywhere continuous and nowhere differentiable function, or fun
ctions, as they more properly are.  These have the form " }{XPPEDIT 
18 0 "Sum(b^n*cos(a^n*Pi*x),n = 0 .. infinity);" "6#-%$SumG6$*&)%\"bG%
\"nG\"\"\"-%$cosG6#*()%\"aGF)F*%#PiGF*%\"xGF*F*/F);\"\"!%)infinityG" }
{TEXT -1 53 "  where 0 < b < 1, a is an odd integer, and ab > 1 + " }
{XPPEDIT 18 0 "3*Pi/2;" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT -1 
62 ".   The proof is now easily accessible from David Bressoud's, " }
{TEXT 260 36 "A Radical Approach to Real Analysis." }{TEXT 262 0 "" }
{TEXT -1 1 " " }{TEXT 261 1 " " }{TEXT -1 47 "Following one of Bressou
d's exercises, we take " }{XPPEDIT 18 0 "Sum((6/7)^n*cos(7^n*Pi*x),n =
 0 .. infinity)" "6#-%$SumG6$*&)*&\"\"'\"\"\"\"\"(!\"\"%\"nGF*-%$cosG6
#*()F+F-F*%#PiGF*%\"xGF*F*/F-;\"\"!%)infinityG" }{TEXT -1 131 " as an \+
example and make our acquaintance with this famous family of functions
 by using Maple to graph its first few partial sums.  " }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 12 "The Function" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "F:=(n,x)->(6/7)^n*cos(7^n*Pi*x):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "Sum(F(n,x),n=0..infinity);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 115 "To get some idea of the graph of this function
, let us inspect the graphs on [-1,1] of the first four partial sums.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "for k from 1 to 3 do\n
P[k]:=plot(sum(F(n,x),n=0..k),x=-1..1,y=-3..3.5,color=blue,thickness=2
):\nT[k]:=textplot([0.6,3,cat(k+1,`  terms`)],font=[TIMES,BOLD,12])\no
d:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(seq(display
(P[k],T[k]),k=1..3),insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Examining onl
y those portions of the graphs on [0, 0.02], we see, for the first six
 partial sums," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "for k fr
om 1 to 5 do\nP[k]:=plot(sum(F(n,x),n=0..k),x=0..0.02,y=-3..4,color=bl
ue):\nT[k]:=textplot([0.015,3,cat(k+1,`  terms`)],font=[TIMES,BOLD,12]
)\nod:\ndisplay(seq(display(P[k],T[k]),k=1..5),insequence=true);\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 366 "So we come easily to the opinion that the graph of the f
unction defined by the full series is very jagged indeed.  Also, the g
raph of the full series may have properties quite different than the g
raphs of the partial sums.  It will be remembered that this is true of
 the circle, which is the limit of a sequence of regular polygons whic
h may be inscribed inside it. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
123 "However, with the help of Maple, we can do more than this.  We ca
n actually explore the key idea which underlies the proof." }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 17 "Differentiability" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 44 "The proof that Weierstrass' function f(x) = " }
{XPPEDIT 18 0 "Sum((6/7)^n*cos(7^n*Pi*x));" "6#-%$SumG6#*&)*&\"\"'\"\"
\"\"\"(!\"\"%\"nGF*-%$cosG6#*()F+F-F*%#PiGF*%\"xGF*F*" }{TEXT -1 79 " \+
 is nowhere differentiable turns on the fact that, given any value of \+
x, say, " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 162 " , t
here is no real number, L, which can be the value of the derivative th
ere.  This is shown by, first, observing that if there were such an L,
 then, given 0 <  " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT 
-1 23 ", there must be a 0 <  " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }
{TEXT -1 12 ", such that " }{XPPEDIT 18 0 "abs((f(x)-f(x[0]))/(x-x[0])
-L) < epsilon;" "6#2-%$absG6#,&*&,&-%\"fG6#%\"xG\"\"\"-F+6#&F-6#\"\"!!
\"\"F.,&F-F.&F-6#F3F4F4F.%\"LGF4%(epsilonG" }{TEXT -1 18 "   whenever \+
  0 < " }{XPPEDIT 18 0 "abs(x-x[0]) < delta;" "6#2-%$absG6#,&%\"xG\"\"
\"&F(6#\"\"!!\"\"%&deltaG" }{TEXT -1 13 " ;  that is, " }{XPPEDIT 18 
0 "abs((f(x)-f(x[0]))/(x-x[0]));" "6#-%$absG6#*&,&-%\"fG6#%\"xG\"\"\"-
F)6#&F+6#\"\"!!\"\"F,,&F+F,&F+6#F1F2F2" }{TEXT -1 3 " < " }{XPPEDIT 
18 0 "abs(L)+epsilon;" "6#,&-%$absG6#%\"LG\"\"\"%(epsilonGF(" }{TEXT 
-1 16 " whenever   0 < " }{XPPEDIT 18 0 "abs(x-x[0]) < delta" "6#2-%$a
bsG6#,&%\"xG\"\"\"&F(6#\"\"!!\"\"%&deltaG" }{TEXT -1 26 ".  Then it is
 shown that  " }{XPPEDIT 18 0 "abs((f(x)-f(x[0]))/(x-x[0]));" "6#-%$ab
sG6#*&,&-%\"fG6#%\"xG\"\"\"-F)6#&F+6#\"\"!!\"\"F,,&F+F,&F+6#F1F2F2" }
{TEXT -1 214 "  is not bounded above, so there can be no such number L
. Of course, it is not possible to sketch the graph of f to illustrate
 this, but the fact can be made plausible by using a sketch of a suita
ble partial sum,  " }{XPPEDIT 18 0 "f[m];" "6#&%\"fG6#%\"mG" }{TEXT 
-1 38 ",  and showing graphically that, once " }{XPPEDIT 18 0 "delta;
" "6#%&deltaG" }{TEXT -1 41 " is fixed, any proposed upper bound for  \+
" }{XPPEDIT 18 0 "abs((f[m](x)-f[m](x[0]))/(x-x[0]));" "6#-%$absG6#*&,
&-&%\"fG6#%\"mG6#%\"xG\"\"\"-&F*6#F,6#&F.6#\"\"!!\"\"F/,&F.F/&F.6#F6F7
F7" }{TEXT -1 47 "  can be exceeded by suitably choosing m and x." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "We do this in the case " }
{XPPEDIT 18 0 "x[0] = 0;" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 116 " and sho
w that for the proposed upper bounds 50, 300, 2,000, 15,000, and 100,0
00, these can be exceeded by choosing " }{TEXT 258 1 "x" }{TEXT -1 23 
" sufficiently close to " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }
{TEXT -1 26 "without needing to choose " }{TEXT 259 1 "m" }{TEXT -1 
31 " larger than 6.  We use f(x) = " }{XPPEDIT 18 0 "Sum((6/7)^n*cos(7
^n*Pi*x),n = 0 .. infinity)" "6#-%$SumG6$*&)*&\"\"'\"\"\"\"\"(!\"\"%\"
nGF*-%$cosG6#*()F+F-F*%#PiGF*%\"xGF*F*/F-;\"\"!%)infinityG" }{TEXT -1 
1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F:=(n,x)->(6/7)^n*cos(
7^n*Pi*x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=(m,x)->sum
(F(n,x),n=0..m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "slope:=
(m,x)->(f(m,x)-f(m,0))/x:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "X:=[-.11,-0.015,-0.0024,-0.00032,-0.000044,-0.0000062]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=[0.5,0.1,0.01,0.001,0.000
2,0.00003]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "M:=[seq(slop
e(k,X[k]),k=1..6)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "M1:=
[0,50,300,2000,15000,100000]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "S:=m->piecewise(m<2,3,M1[m]*x+f(m,0)):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "SL:=m->M[m]*x+f(m,0):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 280 "PIC:=m->plot([f(m,x),SL(m),S(m)],x=-A[m]..A[m],y=
piecewise(m<2,-(f(m,0)+0.5)..f(m,0)+0.5,0..f(m,0)+0.5),color=[blue,gre
en,red],thickness=[2,3,2],title=cat(`green slope = `,convert(round(M[m
]),string),`,  ` , m+1,` terms,`,`  x = `,convert(X[m],string)),titlef
ont=[TIMES,BOLD,12]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "p
ic:=m->plot([f(m,x),S(m+1)-f(m,0)+f(m-1,0)],x=-A[m]..A[m],y=piecewise(
m<2,-(f(m,0)+0.5)..f(m,0)+0.5,0..f(m,0)+0.5),color=[blue,red],thicknes
s=[3,2],title=cat(`slope = `,convert(round(M1[m+1]),string),`,  ` , m+
1,`terms`),titlefont=[TIMES,BOLD,12]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(PIC(1));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(pic(1));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(PIC(2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(pic(2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(PIC(3));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(pic(3));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(PIC(4));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(pic(4));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(PIC(5));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(pic(5));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "display(PIC(6));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The pictures \+
are suggestive.  Now it is time to look at the proof.  " }}}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 10 "References" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 75 "1.  M. Holmes, et. al., Exploring Calculus with Maple, Ad
dison-Wesley, 1993" }}{PARA 0 "" 0 "" {TEXT -1 64 "2.  D. Bressoud, A \+
Radical Approach to Real Analysis, MAA, 1994." }}{PARA 0 "" 0 "" 
{TEXT -1 98 "3.  R. Courant, Differential and Integral Calculus, Inter
science Publishers, Inc., New York, 1937." }}{PARA 0 "" 0 "" {TEXT -1 
152 "4.  B. Riemann, \"Ueber die Darstellbarkeit einer Function durch \+
eine trigonometrische Reihe,\" Gesammelte Mathematische Werke, Leipzig
, 1876, pp.213-253." }}{PARA 0 "" 0 "" {TEXT -1 59 "5.  K. Weierstrass
, Mathematische Werke, vol. 2, pp. 71-74." }}}}{MARK "4" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }