# Convergence

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

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Uniform convergence:

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Pointwise – but not uniform – convergence:

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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