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.