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