Majorant criterion
The majorant criterion is a mathematical convergence criterion for infinite series . The basic idea is to estimate a series by a larger one, called a majorante , whose convergence is known. Conversely, the divergence can be demonstrated with a minor edge .
Formulation of the criterion
Be an infinite series
given with real or complex summands . Now is there a convergent infinite series
with non-negative real terms and applies to almost all :
then the series is absolutely convergent . It is said that the row is majorized by or is the majorante of .
If one reverses this conclusion, one obtains the minorant criterion : are and series with nonnegative real summands or , and applies
for almost everyone , then it follows: If it is divergent, then it is also divergent.
proof
The series converges , then, for every one , so for all (does Cauchy's convergence test ).
From the triangle inequality and it follows . From this follows the (absolute!) Convergence of according to the Cauchy criterion .
example
The geometric series
is convergent. Because of this , the series also converges
- .
Applications
The majorant criterion is also referred to as the most general form of a comparison criterion of the first type, all others result from the insertion of specific series for . Most prominent are the root criterion and the quotient criterion , in which the geometric series is chosen as a comparison series .
Cauchy's compression criterion can also be derived from the majorant or minorant criterion , with which, for example, it can be shown that the harmonic series
convergent for and divergent for is.
The majorant criterion can be extended to the case of normalized vector spaces ; it then states that if it applies to almost all , the partial sum sequence of is a Cauchy sequence . Is the room complete , i.e. H. a Banach space , so converges if converges. In particular, Banach's fixed point theorem follows , which is used in many constructive theorems of analysis.
See also
Web links
literature
- Otto Forster : Analysis 1. Differential and integral calculus of a variable. 8th edition. Vieweg-Verlag, 2006. ISBN 3-8348-0088-0