Let $f$ be a smooth real function defined around origin. If we informally differentiate from the series

$\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$ term by term we get $\frac{d}{dx}\hat{f}(x)=0$. Thus we may conclude that the series is convergent to the constant $f(0)$. But as it is well-known one can not be allowed to differentiate from a series term by term. Now the questions are:

i) Is the series convergent around origin?

ii) If it is convergent then is it convergent to the constant $f(0)$.

In fact one can show that

a) If $f$ is analytic at origin then the series $\hat{f}$ is convergent uniformly to the constant $f(0)$.

b)If $f$ is nowhere analytic in the sense that the radius of convergence of the Taylor's series is zero then of course the series is divergent. But if $f$ is nowhere analytic in the sense that the radius of convergence of the Taylor's series is positive but the Taylor's series does not converge to the function $f$ then the series $\hat{f}$ may converge.

c) About the function $f(x):=e^{\frac{-1}{x^2}},f(0)=0$ one can show that if the series is convergent then its sum is constant.

c) There are nowhere analytic functions such that the series is convergent in a dense subset to the constant $f(0)$ and there are nowhere analytic functions such that the series is divergent everywhere.

Now the main questions are.

Is there a smooth function $f$ which is not analytic at origin and the series $\hat{f}$ is convergent in an interval around origin and the sum is the constant $f(0).$?

Is there a smooth function $f$ which is not analytic at origin and the series $\hat{f}$ is convergent in an interval around origin and the sum is not constant.?

If we define a linear differential operator of infinite order $f\mapsto \hat{f}-f(0)$. Then in above we said that analytic functions at origin are contained in the space of eigenfunctions of the zero eigenvalue of this operator. Now the question arises that: are there nonzero eigenvalues for this operator?

For the function $f(x):=e^{\frac{-1}{x^2}},f(0)=0$, is the series $\hat{f}$ convergent? Please see the preprint arXiv:1105.2611v2 [math.GM] 5 Jun 2011