How can we pick two non-zero functions $f$ and $g$ so that for all integer $k\ge 0$
$$\sum_{n=1}^\infty f(n)g(n)^k = \left(\sum_{n=1}^\infty f(n)g(n) \right)^k$$
Assume $f$, $g$, and $k$ allow for convergence. Let neither $f$ nor $g$ involve $k$.
Note that $k=0$ implies $$\sum_{n=1}^\infty f(n) = 1$$
As @Kavi pointed out, solutions for $f$ are trivial when $g=1$. I would ideally wish for a general set of solutions, but I am definitely interested in any interesting not-obvious solutions.
Note: I made quite a few edits to simplify my question.