# What is the difference between a Taylor series, Taylor polynomial, analytic function and a quadradic approximation?

I am using Wikipedia to brush up on these key concepts. I will write what I believe I read and maybe an expert can put me back on track. First , they are all functions.

I get the idea that a Taylor polynomial is the same as a Taylor series other than maybe the polynomial is finite?

So that maybe a quadradic approximation is only the first two terms of a Taylor series?

And finally all Taylor series are analytic but going the other way I am not sure, since all analytic functions are locally given by a convergent power series.

Would the function having the real numbers as domain and the square as the output be considered "continuously differentiable" and it's Taylor series would have all it's terms 0 accept for the first two?

• I'm not clear which function you're referring to in your final paragraph. Are you referring to the function $x \mapsto x^2$? If so, the function is continuously differentiable, analytic, and is literally its own Taylor series. – Theo Bendit Sep 16 '18 at 20:21

For a given function $f(x)$ and a given point $x=a$, the Taylor polynomial of degree n is the polynomial $$P(x) = f(a) + f'(a) (x-a) + f''(a)(x-a)^2 /{2!}+...+ f^{(n)} (a) (x-a)^n /{n!}$$ This polynomial has its value and up to $n^{th}$ derivatives match with the function $f(x).$
The Taylor series may or may not converge to the function $f(x).$