Question: Is there an example for $\sum a\_nx^n$ with radius $R>0$ such that $a\_n \neq 0$ for some $n$ ($a\_n$ is not always $0$) but for all $x \in (-R,R)$ we have $\sum a\_nx^n=0$

Answer: 
Since the power series $g$ is convergent in the radius, it is infinitely times differentiable there. Since it is uniformly 0 there, it must have each of its derivatives be 0. Therefore, the constant term of $\frac{d^n}{dx^n}g$ must be 0 for all $n$. This implies that $a\_n=0$ for all $n$.
Hint:

Differentiate the power series $n$ times to find $a\_n$.