Comparison Test/Corollary 2

Corollary to Comparison Test
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ be two series of positive real numbers.

Let $\lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = k$ for some $k \in \R$.

Then either:
 * both $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ are convergent

or:
 * both $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ are divergent.