Can the original function be derived from its k t h order Taylor polynomial?

A finite Taylor polynomial certainly cannot determine the function uniquely. For instance, $f(x)=1+x+\frac{x^2}{2}$ and $g(x)=e^x$ have the same second-order Taylor polynomial at $c=0$.
As it turns out, there are functions which cannot be recovered from their Taylor series. For instance, define $f(x)=e^{-\frac{1}{x}}$ if $x>0$ and $f(x)=$ for $x\le 0$. Then $f^{(n)}(0)=0$ for all $n$, but $f$ is not identically zero.