Is there a non zero power series that always converges to $0$?

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 1

Hint:

Differentiate the power series $n$ times to find $a_n$.

Answer 2

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 $n$.

Answer 3

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.