If $\sum a_n/k^n = 0$ for all $k$, then $a_n = 0$ for all $n$

Let $a_n$ be a sequence of complex numbers such that the sum from $n=1$ to infinity of $a_n/k^n$ is zero, for all k=1,2,3,... That is $$\sum_{n=1}^{\infty}\frac{a_n}{k^n}=0, k=1,2,3,...$$ Prove that $a_n=0$ for all n.

I want to use the identity theorem I think (it seems like it might work here), but I can't see how to apply it in this situation. Could anyone help with this please?

Since $\lim_{n\to\infty }\frac{a_n }{2^n } =0 $ we obtain that $|a_n |<2^n $ for large $n .$ Hence the power series $$ G(z) =\sum_{j=0}^{\infty } a_n z^n $$ defines a holomorphic function in the disc $\{z :|z| <\frac{1}{2} \} .$ Hence we can use the identity theorem.