# 9.4 | Comparisons of Series

Direct Comparison Test
Let $$0 < a_n \leq b_n$$ for all $$n$$.
1. If $$\displaystyle \sum_{n = 1}^\infty b_n$$ converges, then $$\displaystyle \sum_{n = 1}^\infty a_n$$ converges
2. If $$\displaystyle \sum_{n = 1}^\infty a_n$$ diverges, then $$\displaystyle \sum_{n = 1}^\infty b_n$$ diverges
Limit Comparison Test
If $$a_n > 0, b_n >0$$, and
$$\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L$$
where $$L$$ is finite and positive, then
$$\sum_{n = 1}^\infty a_n$$
and
$$\sum_{n = 1}^\infty b_n$$
either both converge or both diverge.