Let $M$ be a complex 2-dimensional manifold with the additional property that a smooth function $h:M \to \mathbb C$ is constant if and only if $\partial\overline\partial h = 0$ (which is equivalent (if I'm not mistaken) to $M$ having only the constants as holomorphic functions).
Now we consider a family of smooth functions $f_n : M \to \mathbb C$. We know that $$\sum_{n=1}^\infty \partial\overline\partial f_n$$ converges locally uniformly to a (1,1)-form $g$. Further we know that for one $x_0 \in M$ $$\sum_{n=1}^\infty f_n(x_0)$$ is convergent. Does this imply that $$f(x):=\sum_{n=1}^\infty f_n(x)$$ converges for all $x \in M$ to a smooth function $f$ with $\partial\overline\partial f=g$?