Partial Differential Equations PowerTool

by Dr. Jim Herod

Section 2.1: Convergence of Infinite Sequences and Series

In this section, we address the general 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, and uniform convergence.

     We suppose that we have a sequence of functions  ,  , , ... and another function g ( x ). We say that the  's converge to  g  on the interval [ a , b ] in the sense of

1. norm convergence ,  if   -> 0  as  n ->  .

2. pointwise,  if  for each x in [ a , b ],      -> g(x)  as  n ->  .

3. uniformly,  if the maximum for all x in [ a , b ] of  goes to zero as n  ->    .

As it might be expected these definitions of convergence are not all equivalent. Uniform convergence is the strongest condition, in the sense that if a sequence converges uniformly, then it will converge on C([ a , b ]) in the norm and pointwise, too. When making a model, this strong uniform convergence is to be desired. If that type of convergence does not hold, then one should get the strongest that is available. From what has come before, we know that if we begin with an orthogonal sequence and generate a Fourier Series for a function f , then the Fourier Series will converge in norm.We will try to understand the other two types of convergence in this and following sections.

In order to give an understanding of the definitions of convergence, we compare and contrast them for several sequences of functions.

1. Uniform Convergence implies pointwise convergence . To see this, note only that if  

                                  ->   0

then for each x,         ->   0.

2. Uniform Convergence implies normed convergence.  To see this, note that

                            .

3. Pointwise convergence does not imply uniform convergence.  To see this, note that the following sequence of functions converges to the zero function g(x)=0 pointwise, but the maximum of each function differs from the zero function by

                                             1/e = 0.367689... .

How shall the sequence of functions be displayed? One way is to plot them all.

>	plot([seq(n*x*exp(-n*x),n=1..10)],x=0..1);

An alternate method of displaying the sequence is to show the terms of the sequence 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 frame" command.

>	display([seq(plot(n*x*exp(-n*x),x=0..1),n=1..20)],insequence=true);

>

Finally, I try to convince you that the sequence converges to zero pointwise by choosing one point and letting you observe the passage to nearly zero. I choose x  = 0.8.

>	for n from 1 to 15 do    n*0.8*exp(-n*0.8); end do; n:='n':

4. Pointwise convergence does not imply normed convergence . To see this, take   to be defined this way

                                    =  , if .

We illustrate with graphs that the sequence converges to zero pointwise, but the integral of the difference squared goes to infinity. First, we draw the graphs.

>	display([seq(plot(n^2*x*exp(-n*x),x=0..1),n=1..10)],insequence=true);

To see, for sure, that the sequence converges pointwise, take a positive x and evaluate what happens as n increases without bound

>	assume(x>0); limit(n^2*x*exp(-n*x),n=infinity); x:='x':

Now, to see that the sequence does not converge in norm, we evaluate the integral and let n go to infinity.

>	Int((n^2*x*exp(-n*x))^2,x=0..1)=int((n^2*x*exp(-n*x))^2,x=0..1);

>	limit(%,n=infinity);

>

5. Norm convergence does not imply pointwise convergence . The following is nine terms of an infinite sequence. The sequence does not converge pointwise. To see this, after executing the display  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 do loop . See how the computations might go.

>	with(plots):

>

f[1]:=x->(1+signum(1/2-x))/2: f[2]:=x->(1+signum(x-1/2))/2: f[3]:=x->(1+signum(1/3-x))/2: f[4]:=x->(1+signum(2/3-x))/2-(1+signum(1/3-x))/2: f[5]:=x->(1+signum(1-x))/2-(1+signum(2/3-x))/2: f[6]:=x->(1+signum(1/4-x))/2-(1+signum(0-x))/2: f[7]:=x->(1+signum(1/2-x))/2-(1+signum(1/4-x))/2: f[8]:=x->(1+signum(3/4-x))/2-(1+signum(1/2-x))/2: f[9]:=x->(1+signum(1-x))/2-(1+signum(3/4-x))/2:

>	for n from 1 to 9 do    p[n]:=plot([f[n](x)],x=0..1): end do:

>	display([seq(p[n],n=1..9)],insequence=true);

>	for n from 1 to 9 do int(f[n](x)^2,x=0..1); od;

>

>

We have introduced these notions of convergence in the context of sequences. A series is a sequence. After all, one has only to identify

                            =

to convert these statements of convergence of sequences to statements about convergence of series. In the next Section, we will discuss how series converge with this language.

EMAIL: herod@math.gatech.edu   or   jherod@tds.net

URL: http://www.math.gatech.edu/~herod

Copyright ©  2003  by James V. Herod

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