# Series - Alternating series

1. The Alternating Series Test ( Leibniz's Theorem )
Let $\{a_n \}$ be a sequence of positive numbers such that
$a_{n+1}<a_n$ for all n .
$\displaystyle\lim_{n \to \infty } a_n = 0.$
Then the alternating series $\displaystyle\sum_{n= 1 }^\infty(-1)^na_n$ and $\displaystyle\sum_{n= 1 }^\infty (-1)^{n-1}a_n$ both convergence
2. Absolute Convergence
*   A series $\displaystyle\sum_{n= 1 }^\infty a_n$ is absolutely conveergent if the series
$\displaystyle\sum_{n= 1 }^\infty |a_n|$ is convergent.
*   If the series $\displaystyle\sum_{n= 1 }^\infty a_n$ is absolutely convergent then it is convergent.
3. Conditional convergence
A series $\displaystyle\sum_{n= 1 }^\infty a_n$ is conditionnally convergent if the series is convergent but is not absolutely convergent.