Difference between revisions of "Uniform Convergence of Sequences of Functions"

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<math>\begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^n)_{n=1}^{\infty} = (x, x^2, x^3, ..., x^n, ...) \end{align}</math>
 
<math>\begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^n)_{n=1}^{\infty} = (x, x^2, x^3, ..., x^n, ...) \end{align}</math>
 
<p>This is a sequence of the simplest <math>n^{\mathrm{th}}</math> degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:</p>
 
<p>This is a sequence of the simplest <math>n^{\mathrm{th}}</math> degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:</p>
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" class="image" /></div>
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[http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png Graph of a Sequence of Functions]
  
 
==Pointwise Convergence of Functions==
 
==Pointwise Convergence of Functions==

Revision as of 11:39, 27 October 2021

Sequences of Functions

Definition: An Infinite Sequence of Functions is a sequence of functions with a common domain. The Term of the sequence is the function .

We can define a finite sequence of functions analogously. A finite sequence of functions is denoted .

We can also denote an infinite sequence of functions as simply . We can also use curly brackets to denote a sequence of functions such as or simply .

For example, consider the following sequence of functions:

This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:

<img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png" class="image" />

For another example, consider the following sequence of functions:

This is a sequence of the simplest degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:

Graph of a Sequence of Functions

Pointwise Convergence of Functions

Definition: Let be a sequence of functions with common domain . Then is said to be Pointwise Convergent to the the function written if for all and for all there exists a such that if then .

For example, consider the following sequence of functions defined on :

We claim that is pointwise convergent to . The following image shows the first six functions in the sequence given above. It should be intuitively clear that the sequence converges to the limit function .

To show this, fix and assume that and let be given. Then since we have that:

Choose such that which can be done by the Archimedean property. Then and so for we have that:

Therefore for . Now, for , notice that:

This sequence clearly converges to . So, we conclude that for all . Hence the sequence is pointwise convergent on all of .

Uniform Convergence of Sequences of Functions

Recall from the <a href="/pointwise-convergence-of-sequences-of-functions">Pointwise Convergence of Sequences of Functions</a> page that we say the sequence of functions with common domain is convergent to the limit function if for all and for all there exists an such that if then .

Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!

Definition: Let be a sequence of functions with common domain . Then is said to be Uniformly Convergent to the the limit function (written as uniformly on or as uniformly on ) if for all there exists a such that if then for all .

Graphically, if the sequence of functions are all real-valued and uniformly converge to the limit function , then from the definition above, we see that for all there exists an such that for all we have that the following inequality holds for all :

The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}<math>:

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