How to find some $C^\infty$ functions that do not satisfy the uniqueness theorem for analytic functions

Question

The uniqueness theorem for analytic functions states that suppose two series $\sum_{n=0}^\infty s_nx^n$ and$\sum_{n=0}^\infty t_nx^n$ converges in the interval $(-R,R)$. If the set of $x$ that satisfies $$\sum_{n=0}^\infty s_nx^n=\sum_{n=0}^\infty t_nx^n$$ has a limit point in the interval, then $s_n=t_n$ for all $n \in N$. I know there are lots of functions which are infinitely differentiable but not analytic, such as the one in the wikipedia http://en.wikipedia.org/wiki/Non-analytic_smooth_function, but I do not know how to use these functions to give a counterexample of the uniqueness theorem. Can anyone give me a hint?

Answer 1

You cannot use that to give a counterexample. The theorem assume already that the two series converge in an interval. Thus they are both analytic functions (Not only smooth) on that interval.