Consider the following alternating
series (where ak > 0
for all k) and/or its equivalents.
\[\sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^{k + 1}}{a_k}} = {a_1} - {a_2} + {a_3} - {a_4} + \cdots \]
If the series converges to S, then the nth partial sum Sn and the corresponding remainder Rn can be defined as follows.
\[{S_n} + {R_n} = S\]
\[{S_n} = \sum\limits_{k = 1}^n {{{\left( { - 1} \right)}^{k + 1}}{a_k}} \]
\[{R_n} = \sum\limits_{k = n + 1}^\infty {{{\left( { - 1} \right)}^{k + 1}}{a_k}} \]
This gives us the following
\[{R_n} = S - \sum\limits_{k = 1}^n {{{\left( { - 1} \right)}^{k + 1}}{a_k}} \]
If the series converges to S by
the alternating
series test, then the remainder Rn can be estimated as follows for all n ≥ N:
\[\left| {{R_n}} \right| \le {a_{n + 1}}\]
Note that the alternating series test requires that the numbers a1, a2, a3, ... must eventually be nonincreasing. The number N is the point at which the values of an become non-increasing. an ≥ an +1 for all n ≥ N, where N ≥ 1. |