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What is Cauchy convergence criterion?
Cauchy limit existence criterion, also known as Cauchy convergence principle, gives the necessary and sufficient conditions for convergence.

Cauchy limit existence criterion, also known as Cauchy convergence criterion, is a necessary and sufficient condition (not limited to series) for judging whether a formula converges. Mainly used in the following aspects: series, series of terms, function, generalized integral, series of functions and series of terms of functions correspond to a Cauchy criterion, so the following will explain the criterion according to different aspects.

Extended data:

Improper integral: There are two kinds of improper integral, one is improper integral with infinite integral interval (also called infinite improper integral), and the other is improper integral with unbounded integrand function (also called unbounded improper integral and loss integral). Therefore, there are two corresponding Cauchy convergence criteria. The descriptions of the two criteria are somewhat different, but both can be proved according to the Cauchy convergence criterion of functions.

Function: Considering that the sequence is a special function (that is, the domain is a set of positive integers), we can guess that the convergence and divergence of the function should have a similar conclusion, which is the Cauchy convergence criterion of the function to be said next.

Baidu encyclopedia-Cauchy convergence criterion