What is the difference between pointwise, uniform, and absolute convergence?

[MurraySt]

What is the difference between piecewise, uniform and absolute convergence? When I go about proving whether something converges uniformly vs. just converges do I go about the problem differently? If someone could provide rigorous and layman's terms definitions for these that would be great!

 

[ForMyThunder]

Do you mean "pointwise" instead of "piecewise"?

I assume you know the definition of convergence: for each epsilon, there is a delta... For pointwise convergence, this delta depends on what x you are considering. But for uniform convergence it doesn't depend on the x.

For instance, you might want to search Gibbs Phenomenon. In this case, the functions converge very slowly for points near the jumps in the function while they converge very slowly for those between the jumps. This doesn't happen in uniform convergence; the functions converge at a speed independent of where you look.

Absolute convergence deals with series. It is when the infinite sum of the absolute values convergence.

When you go about proving that a sequence converges, you write delta as a function of epsilon and x, right? But this would prove pointwise convergence. For uniform convergence, you shouldn't have an x so delta should be a function of only epsilon.

 

[HallsofIvy]

Do you mean "pointwise" instead of "piecewise"?

I assume you know the definition of convergence: for each epsilon, there is a delta... For pointwise convergence, this delta depends on what x you are considering. But for uniform convergence it doesn't depend on the x.

For instance, you might want to search Gibbs Phenomenon. In this case, the functions converge very slowly for points near the jumps in the function while they converge very slowly for those between the jumps.

You mean "very fast for those between the jumps".

This doesn't happen in uniform convergence; the functions converge at a speed independent of where you look.

Absolute convergence deals with series. It is when the infinite sum of the absolute values convergence.

When you go about proving that a sequence converges, you write delta as a function of epsilon and x, right? But this would prove pointwise convergence. For uniform convergence, you shouldn't have an x so delta should be a function of only epsilon.