Step By Step Calculus » 17.4 - Alternating Series, Absolute and Conditional Convergence

Synopsis
A series with alternate positive and negative terms is called an Alternating Series. The Alternating Series Test states that an alternating series is convergent if the absolute value of the terms are non-increasing and the limit of the n^{\textrm{th}}n^{\textrm{th}} term goes to zero.
For a convergent alternating series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i, Alternating Series Estimation Theorem states that \displaystyle s_n=\sum_{i=1}^n a_i\displaystyle s_n=\sum_{i=1}^n a_i is an approximate estimate for the sum of the series with an error R_nR_n (also called the remainder of the estimate) where \displaystyle |R_n|\leq |a_{n+1}|\displaystyle |R_n|\leq |a_{n+1}|.
A series \displaystyle\sum_{i=1}^{\infty} a_i\displaystyle\sum_{i=1}^{\infty} a_i is said to be absolutely convergent (or converges absolutely) if the corresponding series of absolute values \displaystyle\sum_{i=1}^{\infty} |a_i|\displaystyle\sum_{i=1}^{\infty} |a_i| is convergent. If a series is absolutely convergent, then it is convergent. But if a series is convergent, it may not be absolutely convergent. A series \displaystyle\sum_{i=1}^{\infty} a_i\displaystyle\sum_{i=1}^{\infty} a_i is said to be conditionally convergent (or converges conditionally) if the series is convergent but not absolutely convergent.