Difference between revisions of "Alternating sum"
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For example, the alternating [[harmonic series]] is <math>1 - \frac12 + \frac13 - \frac 14 + \ldots = \sum_{i = 1}^\infty \frac{(-1)^{i+1}}{i}</math>. | For example, the alternating [[harmonic series]] is <math>1 - \frac12 + \frac13 - \frac 14 + \ldots = \sum_{i = 1}^\infty \frac{(-1)^{i+1}}{i}</math>. | ||
| − | Alternating sums also arise in other cases. For instance, the [[divisibility rule]] for 11 is to take the alternating sum of the [[digit]]s of the [[integer]] in question and check if the result is divisble by 11. | + | Alternating sums also arise in other cases. For instance, the [[divisibility_rules|divisibility rule]] for 11 is to take the alternating sum of the [[digit]]s of the [[integer]] in question and check if the result is divisble by 11. |
Given an [[infinite]] alternating sum, <math>\sum_{i = 0}^\infty (-1)^i a_i</math>, with <math>a_i \geq 0</math>, if corresponding sequence <math>a_0, a_1, a_2, \ldots</math> approaches a [[limit]] of [[zero (constant) | zero]] [[monotonic]]ally then the series converges. | Given an [[infinite]] alternating sum, <math>\sum_{i = 0}^\infty (-1)^i a_i</math>, with <math>a_i \geq 0</math>, if corresponding sequence <math>a_0, a_1, a_2, \ldots</math> approaches a [[limit]] of [[zero (constant) | zero]] [[monotonic]]ally then the series converges. | ||
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Revision as of 21:06, 29 August 2006
An alternating sum is a series of real numbers in which the terms alternate sign.
For example, the alternating harmonic series is 
.
Alternating sums also arise in other cases. For instance, the divisibility rule for 11 is to take the alternating sum of the digits of the integer in question and check if the result is divisble by 11.
Given an infinite alternating sum, 
, with 
, if corresponding sequence 
approaches a limit of zero monotonically then the series converges.
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