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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. 
Arithmetic Series Geometric Series Some Finite Series Infinite Series Properties of Convergent Series Convergence Tests Alternating Series Power Series Differentiation and Integration of Power Series Taylor and Maclaurin Series Binomial Series Fourier Series