Regulate of Cauchy

The rule of Cauchy , which owes its name with the Mathématicien French Augustin Cauchy, is a criterion of convergence for a series with complex terms, or terms in a Espace of Banach.

One gives him sometimes also the name of “criterion of Cauchy” but there is risk of confusion with the very useful statement “all Suite of Cauchy of realities (or complexes) converges” which is that one generally indicates like the “criterion of Cauchy”, and which can also be written for the study of the series.

Simplified statement

Either $(x_n) \,$ a continuation with values in a vector Space normalized, for example a continuation of complexes, and such as the following limit exists:

p= \ lim_ {N \ to + \ infty} \ sqrt {\ left \|x_n \ right \|}

if p is strictly lower than 1 then the continuation $(x_n) \,$ converges towards zero and even the series of general term $(x_n) \,$ is absolutely convergent.

Therefore, in this case, if E is complete (for example if E = C ), the series is convergent.

if p is strictly higher than 1, then the continuation does not tend towards 0, therefore the series diverges coarsely.

if p is worth 1, one cannot conclude: it is the doubtful case rule of Cauchy

It is necessary to take guard with the two following facts:

this limit does not exist inevitably

the rule does not have the reciprocal one: the series of general term $(x_n) \,$ can be absolutely convergent without the continuation of general term $\ sqrt {\ left \|x_n \ right \|}$ is not convergent; all that one can say is that if it is convergent, then its limit cannot be strictly higher than 1 and is thus lower or equal to 1.

More precise statement

One finds the same statement by replacing the concept of limit by that of higher Limite in the definition of p .

See too

Continuation of Cauchy
Rule of Alembert

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