Convergence

his page is a sub-page of our page on Basic properties of functions.

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The sub-pages of this page are:

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Related KMR-pages:

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Other relevant sources of information:

Norm
Topology
Metric
Pointwise convergence
Uniform convergence
Almost everywhere convergence

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Uniform convergence:

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The interactive simulation that created this movie.

Pointwise – but not uniform – convergence:


The interactive simulation that created this movie.

When \, n \rightarrow \infty \, this sequence converges pointwise for each value of \, x , i.e., \, f_n(x) \rightarrow F(x) \, for each \, x . However, the convergence is not uniform, since there is no ”tail-value” \, N = N(\epsilon) \, such that the ”tail” of the sequence \, f_n(x) \, stays within \, \epsilon \, of the pointwise limit function \, F(x) \, for EVERY value of \, n that is greater than \, N . In order for the convergence to be uniform, the tail of the sequence \, f_n(x) \, must stay within the epsilon-band of the limit function \, F(x) \, FOR EVERY VALUE of \, \epsilon > 0 .

The disappearing wave:

Let \, g_n(x) = \dfrac{nx}{e^{nx}} \, be given by the red curve and consider the sequence \, { \{ g_n \} }_{n=1}^{\infty} .

Each function \, g_n \, is continuous at the point \, x = 0 ,
since when \, x \rightarrow 0 \, we have \, \lim\limits_{x \to 0} g_n(x) = 0 = g_n(0) \, for each \, n \, .
When \, x = 1/n \, we have \, g_n(1/n) = 1/e , which is the maximum value of \, g_n .

The sequence of functions \, { \{ g_n \} }_{n=1}^{\infty} behaves like a wave that ”compresses itself” towards the point \, x = 0 \, and threatens to break at this point. Each member function \, g_n \, attains its maximum amplitude of \, 1/e \, at the point \, x = 1/n . The “sequence-wave” passes by each point \, x > 0 \, and then ”dies down” towards amplitude \, 0 \, at this point. Yet the wave never reaches the point \, x = 0 , because at this point it always has the amplitude \, 0 \, since \, g_n(0) = 0 \, for each value of \, n .

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