Cauchy's convergence test

The (Bolzano ) Cauchy criterion (also: convergence principle, [ general ] criterion of Bolzano- Cauchy or convergence criterion of Bolzano- Cauchy ) is a mathematical convergence criterion for sequences and series and of fundamental importance for the analysis. With it can be decided whether a sequence or series of real or complex numbers is convergent or divergent. More generally, the Cauchy criterion also be applied to sequences of elements of a complete metric space or on a series of vectors Banach space. It is named after the French mathematician Augustin Louis Cauchy.

• 2.1 criterion
• 2.2 Examples
• 2.3 proof
• 2.4 generalization

Cauchy criterion for sequences

Criterion

A sequence of real or complex numbers converges to a limit in the real or complex numbers, if there is an index to each, so that the distance between any two sequence elements from this index is less than. Formally, the Cauchy property described by a

Applies.

The Cauchy criterion has a fundamental importance for the analysis. A sequence of real or complex numbers converges to a limit that is if and only if it is a Cauchy sequence. This so-called completeness of the real or complex numbers is a fundamental property of these number ranges.

Example

The sequence of real numbers is recursively

Given being. To show the convergence of this sequence with the Cauchy criterion, we first calculated

Where the last estimate from the triangle inequality

Follows, since the individual sequence elements are limited by. Applying the inequality at times, obtained with

General now applies to

And by repeated application of the triangle inequality and the geometric sum formula

For everyone. Thus the sequence is a Cauchy sequence and hence convergent.

Evidence

The proof of the Cauchy criterion can be done using the theorem of Bolzano - Weierstrass as an axiom for the completeness of the real or complex numbers. Is a Cauchy sequence, then one can find an index, so that

Is for everyone. So the Cauchy sequence by

Limited. The theorem of Bolzano - Weierstrass now states that the sequence has a cluster point. Refers to a subsequence that converges to, arises with

That the limit value of the whole sequence has to be.

Generalization

More generally, the Cauchy- criterion can also be used to study the effects of the convergence of elements of a complete metric space. A sequence of elements converges to a limit in the amount, if

Applies, so if they are a Cauchy sequence with respect to the metric. In a non- complete metric space, the Cauchy criterion is only a necessary condition for the convergence of a sequence, that is a given sequence is not Cauchy sequence, so it diverges.

Cauchy criterion for series

Criterion

A number of

With real or complex summands converges to a limit in the real or complex numbers, if

Applies.

Examples

The series converges, since

If is chosen, which is always possible.

In contrast, the harmonic series diverges, because to Choose, arbitrarily, and then always applies

Evidence

It is verified that the sequence of partial sums

Converges. According to the Cauchy criterion for sequences must therefore exist for each indices, so true. Without loss of generality we can now assume. Then by assumption

And thus converges the partial sums to a threshold value and thus the entire turn.

Generalization

More generally, can the Cauchy criterion for series of vectors from a complete normed space grasp. A series of vectors

If and only converges to a limit in the vector space if

Holds, where the norm of the Banach space is.