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
S.
=
∑
n
=
0
∞
a
n
{\ displaystyle S = \ sum _ {n = 0} ^ {\ infty} a_ {n}}
given with real or complex summands . Now is there a convergent infinite series
a
n
{\ displaystyle a_ {n}}
T
=
∑
n
=
0
∞
b
n
{\ displaystyle T = \ sum _ {n = 0} ^ {\ infty} b_ {n}}
with non-negative real terms and applies to almost all :
b
n
{\ displaystyle b_ {n}}
n
{\ displaystyle n}
|
a
n
|
≤
b
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 .
S.
{\ displaystyle S}
S.
{\ displaystyle S}
T
{\ displaystyle T}
T
{\ displaystyle T}
S.
{\ displaystyle S}
If one reverses this conclusion, one obtains the minorant criterion : are and series with nonnegative real summands or , and applies
S.
{\ displaystyle S}
T
{\ displaystyle T}
a
n
{\ displaystyle a_ {n}}
b
n
{\ displaystyle b_ {n}}
a
n
≥
b
n
{\ displaystyle a_ {n} \ geq b_ {n}}
for almost everyone , then it follows: If it is divergent, then it is also divergent.
n
{\ displaystyle n}
T
{\ displaystyle T}
S.
{\ displaystyle S}
proof
The series converges , then, for every one , so for all (does Cauchy's convergence test ).
T
=
∑
ν
=
0
∞
b
ν
{\ displaystyle T = \ sum _ {\ nu = 0} ^ {\ infty} b _ {\ nu}}
ε
>
0
{\ displaystyle \ varepsilon> 0}
N
∈
N
{\ displaystyle N \ in \ mathbb {N}}
∑
ν
=
n
m
b
ν
<
ε
{\ displaystyle \ sum _ {\ nu = n} ^ {m} b _ {\ nu} <\ varepsilon}
m
≥
n
>
N
{\ displaystyle m \ geq n> N}
From the triangle inequality and it follows . From this follows the (absolute!) Convergence of according to the Cauchy criterion .
|
a
ν
|
≤
b
ν
{\ displaystyle | a _ {\ nu} | \ leq b _ {\ nu}}
|
∑
ν
=
n
m
a
ν
|
≤
∑
ν
=
n
m
|
a
ν
|
≤
∑
ν
=
n
m
b
ν
<
ε
{\ displaystyle {\ Big |} \ sum _ {\ nu = n} ^ {m} a _ {\ nu} {\ Big |} \ leq \ sum _ {\ nu = n} ^ {m} | a _ {\ nu} | \ leq \ sum _ {\ nu = n} ^ {m} b _ {\ nu} <\ varepsilon}
S.
=
∑
ν
=
0
∞
a
ν
{\ displaystyle S = \ sum _ {\ nu = 0} ^ {\ infty} a _ {\ nu}}
example
The geometric series
T
=
∑
n
=
0
∞
1
2
n
=
1
1
+
1
2
+
1
4th
+
1
8th
+
1
16
+
⋯
{\ 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
1
2
n
+
1
≤
1
2
n
{\ displaystyle {\ frac {1} {2 ^ {n} +1}} \ leq {\ frac {1} {2 ^ {n}}}}
S.
=
∑
n
=
0
∞
1
2
n
+
1
=
1
2
+
1
3
+
1
5
+
1
9
+
1
17th
+
⋯
{\ 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 .
T
=
∑
n
=
0
∞
b
n
{\ 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
S.
n
=
∑
k
=
1
n
1
k
α
{\ displaystyle S_ {n} = \ sum _ {k = 1} ^ {n} {\ frac {1} {k ^ {\ alpha}}}}
convergent for and divergent for is.
α
>
1
{\ displaystyle \ alpha> 1}
0
<
α
≤
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.
‖
a
n
‖
≤
b
n
{\ displaystyle \ | a_ {n} \ | \ leq b_ {n}}
n
{\ displaystyle n}
S.
=
∑
n
=
0
∞
a
n
{\ displaystyle S = \ sum _ {n = 0} ^ {\ infty} a_ {n}}
S.
{\ displaystyle S}
T
{\ displaystyle T}
See also
Web links
literature
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">