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: 
No, this is not possible. The series converges uniformly on $(-R\_0, R\_0)$ for any $R\_0 < R$, so the coefficient $a\_n$ gives the $n$th derivative of the function $f(x)=\sum a\_nx^n$ defined by the power series at $x=0$. Therefore, if one of the coefficients $a\_n$ is nonzero then $f^{(n)}(0) = a\_n\neq 0$, so $f$ is nonconstant on any sufficiently small neighborhood of 0.
Hint:

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