# 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

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

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}$