Brandon White

2022-11-24

Can the original function be derived from its ${k}^{th}$ order Taylor polynomial?

Skyler Carney

Expert

A finite Taylor polynomial certainly cannot determine the function uniquely. For instance, $f\left(x\right)=1+x+\frac{{x}^{2}}{2}$ and $g\left(x\right)={e}^{x}$ 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 $f\left(x\right)={e}^{-\frac{1}{x}}$ if $x>0$ and $f\left(x\right)=$ for $x\le 0$. Then ${f}^{\left(n\right)}\left(0\right)=0$ for all $n$, but $f$ is not identically zero.

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