Power series vs. taylor series

[ehilge]

Hey all,
So I have a physics final coming up and I have been reviewing series. I realized that I'm not quite sure on what the differences are between a Taylor series and a power series. From what I think is true, a taylor series is essentially a specific type of power series. Would it be correct to say that a power series is just the sum of a random sequence whereas a Taylor series is also a sum of a sequence but one that can approximate a function. Anything I'm missing here? Is there anything else special about a taylor series?
Thanks for your help!

 

[HallsofIvy]

The Taylor's series of a function is a power series formed in a particular way from that function. Of course, if a power series is equal to a function (not "approximate a function") then that power series is the Taylor's series for the function.

On the other hand, it is possible that the Taylor's series for an infinitely differentiable function is NOT equal to the function at all. An example is
$$f(x)= \left{\begin{array}{c} e^{-1/x^2} if x\ne 0 \\ 0 if x= 0\end{array}\right$$
f is infinitely differentiable and the value of every derivative at x= 0 is 0 so its Taylor's series about 0 is just
$$\sum_n=0^\infty \frac{0}{n!}x^n$$
which, of course, converges to 0 for all x but is equal to f only at x= 0.