9.5 | Alternating Series

Alternating Series Test
Let $$a_n > 0$$. The alternating series
$$\sum_{n=1}^\infty (-1)^n a_n$$
and
$$\sum_{n = 1}^\infty (-1)^{n+1} a_n$$
converge when the two conditions listed below are met.

1. $$\displaystyle \lim_{n \rightarrow \infty} a_n = 0$$
2. $$a_{n+1} \leq a_n$$, for all $$n$$
Absolute Convergence
If the series $$\sum |a_n|$$ converges, then the series $$\sum a_n$$ also converges.
Definitions of Absolute and Conditional Convergence

1. The series $$\sum a_n$$ is absolutely convergent when $$\sum|a_n|$$ converges.
2. The series $$\sum a_n$$ is conditionally convergent when $$\sum a_n$$ converges but $$\sum |a_n|$$ diverges.