Let $(a_k)_{k \in \mathbb{N}}$ be a sequence of real numbers. If $\forall n \in \mathbb{N}$: $\sum_{k=0}^{+ \infty} a_k k^n = 0$. Is it true that $a_k=0$ for all $k \in \mathbb{N}$ ?
It looks like a problem of orthogonality in an Hilbert space, but I am not sure here because the sequence $k^n$ is not convergent.