Thanks! If I understand you correctly the dominated "function" in this case is defined by $g_{N,n} = a_n/(b_n+\sigma/N)$ for $n \leq N$ and $g_{N,n} = 0$ for $n > N$. Then $\sum_{n=1}^\infty g_{N,n} = f^N(\sigma)$ but also $g_{N,n} \to a_n/b_n$ pointwise. Since $g_{N,n}$ is dominated by $a_n/b_n$ the convergence follows by dominated convergence wrt to the counting measure. Very helpful!