Clarifications regarding definitions of Taylor, Laurent etc. series

[Avichal]

I want to know the difference between various kinds of series like Taylor, Laurent and Asymptotic.

I have some understanding but I want some clarifications. Here is what I understand:-
1) Taylor series is just f(0) + x.f'(0) + x².f''(0)/2! + ...
2) Laurent series is applied when taylor series cannot be applied like when f(0) is not defined.
3) Asymptotic series is around infinty point.

Am I correct? Correct me if I'm wrong

 

[csopi]

Here are some crucial differences:
1. A Laurent series is convergent on an annulus in the complex plane, whereas the Taylor series is on a disk.

2. EVERY function that is holomorphic on r < | z - a | < R has a unique, convergent Laurent series (defined on this disk. The exact values of the coefficients depend on the annulus). This is a very strong claim, this is far from being true for a Taylor series. You might have R = infinity and r=0.

3. From a practical point of view: a Laurent series contains in general terms with negative powers, e.g. (z-a)^(-n) (when r > 0).

 

[csopi]

Here are some crucial differences:
1. A Laurent series is convergent on an annulus in the complex plane, whereas the Taylor series is on a disk.

2. EVERY function that is holomorphic on r < | z - a | < R has a unique, convergent Laurent series (defined on this disk. The exact values of the coefficients depend on the annulus). This is a very strong claim, this is far from being true for a Taylor series. You might have R = infinity and r=0.

3. From a practical point of view: a Laurent series contains in general terms with negative powers, e.g. (z-a)^(-n) (when r > 0).

 

[Avichal]

Before that I have one more confusion. What does around a point mean when finding series of some function.
When finding taylor series, that means we use derivative at that point to approximate.

In case of laurent series, what does 'around some point' mean?
Also suppose I want a series expansion of e^1/x what would I apply? Laurent or asymptotic?

Also am I write about asymptotic series i.e. it is series around point = infinity?