Comparison test

The majorant criterion is a mathematical convergence criterion for infinite series. The basic idea is a number by a larger, so-called majorant estimate their convergence is known. Conversely, it can be demonstrated with a minorant divergence.

Formulation of the criterion

Be an infinite series

Given real or complex summands. Is there now a convergent infinite series

With non-negative real summands and is valid for almost all n:

Then the S series is absolutely convergent. We say that the series S is majorized by T or T is the majorant of S.

One hand, this conclusion in order to obtain the minorant criterion: Are S and T series with non-negative real summands respectively, and applies

For almost all n, then it follows: If T is time divergent, then S is divergent.

Evidence

The series converges, then there exists for each, such that for all valid ( Cauchykriterium ).

It follows from. It follows that ( absolute! ) convergence of after Cauchykriterium.

Example

The geometric series

Is convergent. Because thus also converges, the series

Applications

The majorant criterion is also referred to as the most general form of a comparison criterion first type, all others are obtained by the insertion of specific rows for. Most prominently are the root test and the ratio test, in which the geometric series is chosen as the reference series.

Also can be derived from the majorant or minorant criterion the Cauchy compression criterion by which it can be shown, for example, that the harmonic series

Convergent and divergent is for.

The majorant criterion can be extended to the case of normed vector spaces, it then states that if for almost all n, the sequence of partial sums of a Cauchy sequence. If the room is full, that is, a Banach space, S converges if T converges. In particular, it follows the fixed point theorem of Banach, which is used in many constructive sets of analysis.