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

Brandon White

Brandon White

Answered

2022-11-24

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

Answer & Explanation

Skyler Carney

Skyler Carney

Expert

2022-11-25Added 9 answers

A finite Taylor polynomial certainly cannot determine the function uniquely. For instance, f ( x ) = 1 + x + 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 1 x if x > 0 and f ( x ) = for x 0. Then f ( n ) ( 0 ) = 0 for all n, but f is not identically zero.

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