Limit Comparison Test

Theorem
Let $$\left \langle {a_n} \right \rangle$$ and $$\left \langle {b_n} \right \rangle$$ be sequences in $\R$.

Let $$\frac {a_n}{b_n} \to l$$ as $$n \to \infty$$ where $$l \ne 0$$.

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

Proof

 * Suppose $$\sum_{n=1}^\infty b_n$$ is convergent.

Then by Terms in Convergent Series Converge to Zero, $$\left \langle {b_n} \right \rangle$$ converges to zero.

A Convergent Sequence is Bounded.

So it follows that $$\exists H: \forall n \in \N^*: a_n \le H b_n$$.

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


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

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