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