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| {{Distinguish|Cauchy condensation test}}
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| The '''Cauchy convergence test''' is a method used to test [[Series (mathematics)|infinite series]] for [[convergent series|convergence]]. A series
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| :<math>\sum_{i=0}^\infty a_i</math>
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| with [[real number|real]] or [[complex number|complex]] summands ''a''<sub>''i''</sub> is convergent if and only if for every <math>\varepsilon>0</math> there is a [[natural number]] ''N'' such that
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| :<math>|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon</math>
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| holds for all {{nobreak|''n'' > ''N''}} and {{nobreak|''p'' ≥ 1}}.<ref>Abbott, Stephen (2001). ''Understanding Analysis'', p.63. Springer, New York. ISBN 9781441928665</ref>
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| The test works because the space '''R''' of real numbers and the space '''C''' of complex numbers (with the metric given by the absolute value) are both [[complete metric space|complete]], so that the series is convergent [[if and only if]] the partial sum
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| : <math>s_n:=\sum_{i=0}^n a_i</math>
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| is a [[Cauchy sequence]]: for every <math>\varepsilon>0</math> there is a number ''N'', such that for all ''n'', ''m'' > ''N'' holds | |
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| <math>|s_m-s_n|<\varepsilon.</math>
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| We can assume ''m'' > ''n'' and thus set ''p'' = ''m'' − ''n''.
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| :<math>|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon.</math>
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| {{PlanetMath attribution|id=3894|title=Cauchy criterion for convergence}}
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Cauchy_criteria Cauchy criteria] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| [[Category:Convergence tests]]
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Latest revision as of 03:16, 7 December 2014
Hi there, I am Sophia. I am an invoicing officer and I'll be promoted quickly. North Carolina is the location he enjoys most but now he is contemplating other options. To climb is some thing I truly appreciate doing.
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