Consider the following alternating
series (where *a*_{k} > 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 *S*_{n} and the corresponding remainder *R*_{n} 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 *R*_{n} 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 *a*_{1}, *a*_{2}, *a*_{3}, ... must eventually be nonincreasing. The number *N* is the point at which the values of *a*_{n} become non-increasing. *a*_{n} ≥ *a*_{n +1} for all *n* ≥ *N*, where *N* ≥ 1. |