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.

E 9.4 Exercises