Dilution (equation)

2014-01-24T15:37:19Z

129.81.28.123: changed increasing solute concentration to decreasing solute concentration in solution

{{Distinguish|Cauchy condensation test}}

The '''Cauchy convergence test''' is a method used to test [[Series (mathematics)|infinite series]] for [[convergent series|convergence]]. A series

:<math>\sum_{i=0}^\infty a_i</math>

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

:<math>|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon</math>

holds for all {{nobreak|''n'' > ''N''}} and {{nobreak|''p'' ≥ 1}}.<ref>Abbott, Stephen (2001). ''Understanding Analysis'', p.63. Springer, New York. ISBN 9781441928665</ref>

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

: <math>s_n:=\sum_{i=0}^n a_i</math>

is a [[Cauchy sequence]]: for every <math>\varepsilon>0</math> there is a number ''N'', such that for all ''n'', ''m'' > ''N'' holds

<math>|s_m-s_n|<\varepsilon.</math>

We can assume ''m'' > ''n'' and thus set ''p'' = ''m'' − ''n''.

:<math>|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon.</math>

{{PlanetMath attribution|id=3894|title=Cauchy criterion for convergence}}

== References ==
{{reflist}}
==External links==
* [http://www.encyclopediaofmath.org/index.php/Cauchy_criteria Cauchy criteria] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

[[Category:Convergence tests]]

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Dilution (equation)