Can any continuous function be represented as an infinite polynomial?
Motivation: the antiderivative
$
\int^\ e^{-x^2}dx\
$
can be expressed as an infinite polynomial(write Taylor series for integrand function and integrate) but this antiderivative has no closed/elementary form expression according to Liouville's theorem but is clearly continuous. So are the rest of the non-elementary functions expressible as infinite polynomials? Fascinating.Any insights on how to proceed????
Best Answer
No!
The functions that are given by a convergent power series are really quite rare in the whole scheme of things. They are called analytic functions.
There are a whole class of functions that are called flat functions. These have all of their derivatives zero at a given point and so, as far as Taylor series can tell, are identically zero. The classical example of a flat function is $x \mapsto \operatorname{e}^{-1/x^2}$. Where $0 \mapsto 0$. In this case all of the derivatives are zero at zero (you have to take limits) and so, as far as Taylor series are concerned, this is the zero function.
In addition, some Taylor series only hold in cetain regions. For example, the Taylor series of $(1-x)^{-1}$ is given by $1+x+x^2+x^3+\cdots+x^k+\cdots$. This is fine for all $-1 < x < 1$, but when $|x|>1$ we have serious trouble.