Since $\sum_n a_n/b_n \sum_ {n=1}^\infty \frac{a_n}{b_n } < \infty$ and $a_n/b_n 0 \ge0$, it follows le \frac{a_n}{b_n + \sigma/N}\le \frac{a_n}{b_n} $, we have that$f_N(\sigma)\to \sum_{n=1}^N \frac{a_n}{b_n + \sigma/N} \to \sum_ {n=1}^\infty a_n/b_n$\frac{a_n}{b_n }$ as $N\to\infty$, just by dominated convergence.
Since $\sum_n a_n/b_n < \infty$ and $a_n/b_n \ge0$, it follows that $f_N(\sigma)\to \sum_ {n=1}^\infty a_n/b_n$ just by dominated convergence.