Limit Comparison Test

Theorem
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$.

Let $\ds \frac {a_n} {b_n} \to l$ as $n \to \infty$ where $l \in \R_{>0}$.

Then the series $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ are either both convergent or both divergent.

Proof
Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be convergent.

Then by Terms in Convergent Series Converge to Zero, $\sequence {b_n}$ converges to zero.

A Convergent Sequence is Bounded.

So it follows that:
 * $\exists H: \forall n \in \N_{>0}: a_n \le H b_n$

Thus, by the corollary to the Comparison Test, $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent.

Since $l > 0$, from Sequence Converges to Within Half Limit:
 * $\exists N: \forall n > N: a_n > \dfrac 1 2 l b_n$

Hence the convergence of $\ds \sum_{n \mathop = 1}^\infty a_n$ implies the convergence of $\ds \sum_{n \mathop = 1}^\infty b_n$.