Precise differences in meaning of Power Series, Taylor Series Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern.  Are these terms be considered interchangeable by mathematicians?  If not, just what is the difference?   Which term to prefer when writing a paper - what should determine the choice?
 A: Power series are polynoms without any further idea of what to and where they fit to.
Taylor series are polynoms created by selecting some point on the function and made from it's derivatives at that point, which means that these series describe the behavior of the function around the point.
Simply: Power series are often Taylor series with zero as selected point.
A: A power series (in the variable $x$, with center $c$) is an infinite series of the form $\sum_{n=0}^\infty a^n(x-c)^n$.
Given a function $f(x)$ and a real number $c$, the Taylor series of $f(x)$ at c is a certain power series in $x$ with center $c$.
So they are not interchangeable. I do not think you should say "the power series of $f(x)$ at $c$," for example.
A: You can talk about the power series expansion of a function at a point or the Taylor series of the function at that point. They mean exactly the same thing in that context. Maybe you should mention that actual context in which your question is arising, to get a clearer answer.
