Alternating Series Remainder
A quantity that measures how accurately the nth partial sum of an alternating series estimates the sum of the series. If an alternating series is not convergent then the remainder is not a finite number.
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. |
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