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

${\displaystyle S=\sum _{n=0}^{\infty }a_{n}}$

given with real or complex summands . Now is there a convergent infinite series ${\displaystyle a_{n}}$

${\displaystyle T=\sum _{n=0}^{\infty }b_{n}}$

with non-negative real terms and applies to almost all : ${\displaystyle b_{n}}$ ${\displaystyle n}$

${\displaystyle |a_{n}|\leq b_{n},}$

then the series is absolutely convergent . It is said that the row is majorized by or is the majorante of . ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle S}$

If one reverses this conclusion, one obtains the minorant criterion : are and series with nonnegative real summands or , and applies ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle a_{n}}$ ${\displaystyle b_{n}}$

${\displaystyle a_{n}\geq b_{n}}$

for almost everyone , then it follows: If it is divergent, then it is also divergent. ${\displaystyle n}$ ${\displaystyle T}$ ${\displaystyle S}$

proof

The series converges , then, for every one , so for all (does Cauchy's convergence test ). ${\displaystyle T=\sum _{\nu =0}^{\infty }b_{\nu }}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle \sum _{\nu =n}^{m}b_{\nu }<\varepsilon }$ ${\displaystyle m\geq n>N}$

From the triangle inequality and it follows . From this follows the (absolute!) Convergence of according to the Cauchy criterion . ${\displaystyle |a_{\nu }|\leq b_{\nu }}$ ${\displaystyle {\Big |}\sum _{\nu =n}^{m}a_{\nu }{\Big |}\leq \sum _{\nu =n}^{m}|a_{\nu }|\leq \sum _{\nu =n}^{m}b_{\nu }<\varepsilon }$ ${\displaystyle S=\sum _{\nu =0}^{\infty }a_{\nu }}$

example

The geometric series

${\displaystyle T=\sum _{n=0}^{\infty }{\frac {1}{2^{n}}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\dotsb }$

is convergent. Because of this , the series also converges ${\displaystyle {\frac {1}{2^{n}+1}}\leq {\frac {1}{2^{n}}}}$

${\displaystyle S=\sum _{n=0}^{\infty }{\frac {1}{2^{n}+1}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{9}}+{\frac {1}{17}}+\dotsb }$.

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 . ${\displaystyle T=\sum _{n=0}^{\infty }b_{n}}$

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

${\displaystyle S_{n}=\sum _{k=1}^{n}{\frac {1}{k^{\alpha }}}}$

convergent for and divergent for is. ${\displaystyle \alpha >1}$ ${\displaystyle 0<\alpha \leq 1}$

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. ${\displaystyle \|a_{n}\|\leq b_{n}}$ ${\displaystyle n}$ ${\displaystyle S=\sum _{n=0}^{\infty }a_{n}}$ ${\displaystyle S}$ ${\displaystyle T}$

See also

• Weierstraß's majorant criterion

Web links

Wikibooks: Math for non-freaks: major and minor criterion  - learning and teaching materials

literature

• Otto Forster : Analysis 1. Differential and integral calculus of a variable. 8th edition. Vieweg-Verlag, 2006. ISBN 3-8348-0088-0