Physical significance of Taylor and Maclaurin series - What is the significance of defining a Maclaurin series in Mathematical Physics? In physics, usually Taylor series is used to express a quantity which keep changes with coordinate. For example the potential energy of a molecule changes with coordinate, so we express the potential energy as a Taylor expansion with respect to the coordinate about the equilibrium configuration. 
I am confusing this with the Maclaurin series in mathematics. If the equilibrium position is zero, it is Maclaurin series. We have the freedom to fix the equilibrium position at zero. In short, what is the physical significance of Maclaurin series? or else, what is the need of defining a Maclaurin series? Any physical interests on Maclaurin series?
One more question I would like to add. If a quantity is changing with respect to its coordinate and doesn't have an equilibrium at any of its configuration, then how we can express it? Taylor series can be applied only if the quantity is changing about an equilibrium configuration?
 A: I do not think Maclaurin series has any special physical significance that is different from the Taylor series. But then for some functions it is desirable to have a series expansion at the origin even though it is singular at the origin. Maclaurin series requires that the function is analytic everywhere within the circle of convergence so it can not be used, that is where the Taylor series which can do the expansion about any point in is used. 
A: If I had to describe the purpose of these series as succinctly as possible it would be these three words: to handle complexity.
What do I mean by this? Lets suppose we need to describe some function $\mathbb{R}^M\to\mathbb{R}^N$. The cardinality of the set of all such functions is $2^{\beth_1} = 2^{\left(2^{\aleph_0}\right)}$: the cardinality of the powerset of the continuum (and thus, by Cantor's theorem, a strictly bigger cardinality than that of the continuum).
In contrast, the set of all statements in a finitely generated languages (English, German, Mandarin .... mathematics as done by the inhabitants on this planet at least) is countable: the cardinality of all possible descriptions in such a language is $\aleph_0$.
So, in restricting ourselves to Taylor series defined functions, we are taming the collection of all possible functions to something that our finitely generated languages can describe. Our human descriptive powers fail long before specifying something out of a set of cardinality $2^{\left(2^{\aleph_0}\right)}$. It's much easier to think of and describe a countable, discrete list of co-efficients than a general function $\mathbb{R}^M\to\mathbb{R}^N$.
We are also heeding the experimental evidence that many things seem to vary smoothly. Smooth isn't quite the same as analytic (defined by a Taylor series), but analyticity given smoothness doesn't seem to be an unreasonable model of this observed smoothness.
The significance of Maclaurin and Taylor series when defining fields or other functions of co-ordinates to physics is the same: the former being a translated, "canonical" version of the latter (we can redefine our co-ordinates to put an "origin" at any point in question).
