sum of functions have essential singularity

Question

It's my exercise.

If $f_k(z)$'s are holomorphic functions on $|z|<ϵ$.

$g(z)$ is holomorphic function on $0<|z|<ϵ$ and has essential singularity at $z=0$.

Suppose that $\sum_{k=1}^{\infty}f_k(z)g(z)^k≡0$ on $|z|<ϵ$. (i.e. $\sum_{k=1}^{\infty}f_k(z)g(z)^k$ are convergent on $0<|z|<ϵ$ and equal to $0$. So $z=0$ is removable singularity.)

Then $f_k(z)$'s are zero functions.

I think it is true, but I can't prove it.

Thank you.

Answer 1

It's not true in general.

Say $g(z)=e^{1/z}$. Let $f_1=1$, $f_2=-1/g$ and $f_k=0$ for $k>2$.