The Weierstrass M-test states
Suppose $(f_n)$ is a sequence of complex-valued functions defined on a set $A$ and that there is a sequence of non-negative numbers $(M_n)$ such that $|f_n(z)|\le M_n$ for all $n\ge 1$ and all $z\in A$ and such that $\sum_{n=1}^\infty M_n$ converges. Then the series $\sum_{n=1}^\infty f_n(z)$ converges uniformly on $A$.
Question: Suppose that we add the hypothesis that $\sum_{n=1}^\infty f_n(z)$ converges absolutely on $A$. Taking $M_n=f_n(z)$, it seems that the Weierstrass M-test implies that each absolutely convergent series is uniformly convergent. Where is the error in this reasoning?