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alternating series remainder

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 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 nN:

\[\left| {{R_n}} \right| \le {a_{n + 1}}\]

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.

anan +1 for all nN, where N ≥ 1.

 

See also

Remainder of a series, convergence tests, divergent series

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