Uniform vs. Non-uniform Convergence

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Arlo Caine, Created with GeoGebra

A Pointwise but not Uniformly Convergent Sequence of Functions

The applet at left will help you to explore non-uniform convergence in a particular example. Let

The sequence {f_1,f_2,f_3,...} of functions converges point-wise to the function f, but not uniformly.

  • Move the slider marked n to see the graph of the nth term of this sequence.
  • Click on the check box to show the limit function.
  • Click the checkbox to show a test point. Move the slider TestPoint to change the test point, and then adjust the slider for n to observe convergence of the sequence of points.
  • Click the check box to display the epsilon tube about the limit function. Notice for each adjustment of epsilon, you cannot make the slider for n large enough to guarantee that the graph of the term of sequence lies completely within the tube.

A Uniformly Convergent Sequence of Functions

The applet at right will help you to explore non-uniform convergence in a particular example. Let

The sequence {f_1,f_2,f_3,...} of functions converges uniformly to f.

  • Move the slider marked n to see the graph of the nth term of this sequence.
  • Click on the check box to show the limit function.
  • Click the check box to display the epsilon tube about the limit function. Notice for each adjustment of epsilon, you can always make the slider for n large enough to guarantee that the graphs of the nth and future terms of sequence lie completely within the tube.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Arlo Caine, Created with GeoGebra

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Arlo Caine, Created with GeoGebra

A Pointwise by not Uniformly Convergent Series

This applet helps you examine the non-uniform convergence of a particular series. Remember that a series converges uniformly on a set S if and only if its sequence of partial sums converges uniformly on S.

The series converges pointwise to f on R but not uniformly on R.

  • Move the slider for n to evolve the sequence of partial sum functions for this series and observe the pointwise convergence to the purple limit function f.
  • Click on the check box to show a moveable test point you can use to analyze the point wise convergence of this series.
  • By adjusting the slider for epsilon, you can change the size of the epsilon tube about the limit function. Note that for small epsilon, you can never make n large enough to ensure that the graph of the nth partial sum function and all future partial sum functions lie completely within the epsilon tube.

A Uniformly Convergent Series of Functions

The applet at right illustrates the convergence of the series

The convergence is uniform, but the sum function is VERY COMPLICATED. In a course on Real Analysis, you would learn how to prove that the sum function of this series is continuous at each point but nowhere differentiable!

  • Move the slider marked n to see the graph of the nth partial sum of this sequence.

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Arlo Caine, Created with GeoGebra