I am once again adding notes

20241030125245.md

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+# What is the Cauchy criterion?
+The Cauchy criterion states, that a series [[20241021151020]] $\sum_{n>=0}a_n$ is convergent [[20241002213119]] only if $\forall \epsilon >0: \exists N(\epsilon)$, so that $|\sum_{k=n}^m a_n|<\epsilon$
+for all $m≥n>N(\epsilon)$
+
+#math #analysis #series

notes/20241021152116.md

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-# What is the majorant criteria for infinite series?
+# What is the majorant criterion for infinite series?
 The majorant criteria states, that if there are two series' [[20241021151020]] $\sum_{n} a_n$ and $\sum_{n} b_n$ and $|a_n| <= b_n$, then $a_n$ is convergent, if $b_n$ is convergent. 
 $b_n$ is called the majorant of $a_n$
 

notes/20241021164907.md

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-# What is the Cauchy Criterium?
+# What is the Cauchy criterion?
 ## Sequence
 The Cauchy Criterium for sequences states, that a real sequence [[20241002211453]] is convergent iff it is a cauchy sequence [[20241021164431]].
 

notes/20241021165339.md

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-# What is the minorant criterium? 
+# What is the minorant criterion? 
 The minorant criterium states, that for two series [[20241021151020]] $\sum_n a_n$ and $\sum_n b_n$, that fulfill $0 <= a_n <= b_n$ for almost all $n$, if $a_n$ is divergent, then $b_n$ is divergent [[20241002213340]].
 
 #math #analysis #series

notes/20241030120213.md

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+# What is a relation? 
+A relation R between two sets[[20240929155814]]  $A$ and $B$ is a subset [[20241002205951]] of their cartesian product $A \times B$. Is $A=B$, then it is called a binary relation.
+You can write $aRb$ instead of $(a,b) \in R$.
+
+#math #settheory #set #adm #relations

notes/20241030120621.md

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+# What is an equivalence relation?
+An equivalence relation on a set [[20240929155814]] $A$  is a relation [[20241030120213]], that fulfills the 
+following conditions:
+1. Reflexivity $\forall a \in A: aRa$
+2. Symmetry $\forall a,b \in A: aRb \implies bRa$
+3. Transitivity $\forall a,b,c \in A: aRb \land bRc \implies aRc$
+
+#math #settheory #relations #adm

notes/20241030121034.md

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+# What is a partial order? 
+A partial order of a set[[20240929155814]] A is a relation [[20241030120213]], that fulfills the following conditions: 
+1. Reflexivity: $\forall a \in A: aRa$
+2. Antisymmetry: $\forall a,b \in A: aRb \not\implies bRa$
+3. Transitivity: $\forall a,b,c \in A: aRb \land bRc \implies aRc$
+
+#math #adm #set #relation #settheory

notes/20241030121415.md

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+# What is a total order? 
+A total order on a set $A$ [[20240929155814]] is a relation [[20241030120213]], that is a special case of the partial order [[20241030121034]].
+It has to fulfill the following extra condition:
+$\forall a,b \in A: aRb \lor bRa$ (all elements need to be comparable to oneanother)
+
+#math #adm #settheory #set #relation

notes/20241030121716.md

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+# What is a partition of a set?
+A partition of a set $A$ [[20240929155814]] is a system of non-empty sets, that fulfill the following property
+
+$\forall i,j \in I: A_i \cap A_j = \empty \land \bigcup_{i\in I} A_i = A$
+
+#math #set #settheory #adm

notes/20241030122128.md

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+# How are partitions related to equivalence relations? 
+An equivalence relation [[20241030120213]] $R$ on a set [[20240929155814]] $A$ partitions A [[20241030121716]] into 
+$A_0,...,A_i$ subsets.
+
+#math #settheory #set #relation #adm

notes/20241030122408.md

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+# What is an equivalence class? 
+An equivalence class is the abstraction of a all elemets of a partition [[20241030121716]] of a set [[20240929155814]] $A_i$, that is a subset of a set $A$ 
+on which an equivalence relation $R$ [[20241030120621]] was applied.
+
+#math #set #relation #settheory #adm

notes/20241030122745.md

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+# What is a power set? 
+A power set $P(A)$ of a set $A$ [[20240929155814]] is a set containing all subsets of $A$.
+
+#math #settheory #set #adm

notes/20241030122947.md

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+# What is a Hasse diagram?
+A Hasse diagram is a way of representing partial orders [[20241030121034]] graphically
+
+#math #relation #adm 

notes/20241030123044.md

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+# What is transfinite induction? 
+Transfinite induction is a proof technique, that is an abstraction of complete induction, focussing more on relations [[20241030120213]], specifically partial orders [[20241030121034]].
+It works by proving the following for a relation $R: R \subset A\times A$ on a set $A$:
+$a\in A \land \forall b\in A: bRa \land P(b) \implies P(a)$
+
+#math #adm #settheory #set #relation

notes/20241030123628.md

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+# When is a series absotely convergent? 
+A series [[20241021151020]] $\sum_{n>=0} a_n$ is called "absolutely convergent", if $\sum_{n>=0}|a_n|$ is convergent [[20241002213119]].
+A aboslutely convergent series is also convergent.
+
+#math #analysis #sequence

notes/20241030124126.md

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+# How are absolute convergence and the majorant criterion related?
+The majorant criterium [[20241021152116]] and aboslute convergence [[20241030123628]] are related in that the majorant constrains the 
+absolutely convergent series [[20241021151020]]. This implies, that the conditionally convergent series is also convergent.
+
+#math #analysis #series

notes/20241030124353.md

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+# When is a series conditionally convergent?
+A series[[20241021151020]] is called conditionally convergent if it is not absolutely convergent [[20241030123628]].
+
+#math #analysis #series

notes/20241030124521.md

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+# What is the root criterion?
+The root criterion specifies, that if $\overline{\lim_{n \to \infty}} \sqrt[n]{|a_n|}<1$, then a series is convergent [[20241002213119]]. Else it is divergent [[20241002213340]].
+
+#math #analysis #series

notes/20241030124817.md

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+# What is the quotient criterion?
+The quotient criterion states, that if a series [[20241021151020]] $a_n$ fulfills $\overline{\lim_{n \to \infty}} |\frac{a_{n+1}}{a_n}|<1$, then it is convergent [[20241002213119]]. Else it is divergent [[20241002213340]].
+
+#math #analysis #series

notes/20241030125015.md

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+# What is the Leibnitz criterion?
+The Leibnitz criterion states, that an alternating series  $\sum_{n>=0} (-1)^n a_n$ [[20241021151020]] is convergent[[20241002213119]]  if $a_n$ is strictly monotonously decreasing.
+
+#math #analysis #series

notes/20241104082758.md

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+# What is the Cauchyproduct of two series'?
+The Cauchyproduct is the product of two series' [[20241021151020]] $\sum_{n>=0}a_n$ and $\sum_{n>=0} b_n$.
+It is written as this:
+$\sum_{n>=0}(\sum_{k=0}^{n}a_k )b_{n-k}$
+
+#math #series #analysis

notes/20241104083323.md

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+# What rules are there for arithmetics with series'?
+There are two rules for arithmetics with convergent[[20241002213119]] series' 
+
+## Addition of convergent series $\sum_{n>=0}a_n$ and $\sum_{n>=0} b_n$
+Two convergent series can be added.
+$\sum_{n>=0} (a_n + b_n) \iff \sum_{n>=0} a_n + \sum_n{n>=0} b_n$
+
+## Multiplying with a scalar factor $\sum_{n>=0} \lambda a_n$
+A series may be multiplied with a scalar factor $\lambda$.
+$\sum_{n>=0} \lambda a_n \iff \lambda \sum_{n>=0} a_n$
+
+#math #series #analysis
+

notes/20241104083948.md

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+# What is a power series? 
+A powerseries is a series [[20241021151020]], that fits a certain build type.
+$\sum_{n>=0} a_n (x-x_0)^n$.
+The factors $a_n$ are called the coefficients while $x_0$ is called the development point.
+
+$\sum_{n>=0}a_n(x)^n$ is also a powerseries with an $x_0$ of 0.
+
+#math #analysis #series

notes/20241104084305.md

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+# When does a power series converge? What is a convergence radius? 
+A power series [[20241104083948]] converges[[20241002213119]] if there is a number $R$ ($0<=R<=\infty$) so that $\forall x \in \mathbb{C} |x-x_0| < R$.
+
+This number $R$ is called the convergence-radius. Everything smaller than $R$ is absolutely convergent and everything larger is divergent.
+There are however the edgecases of $|x-x_0| = R$ which have to be treated seperately.
+$R$ can be calculated like this (1 divided by the lim sup of the root criterion?):
+$R= \frac{1}{\overline{\lim_{n \to \infty}} \sqrt[n]{|a_n|}}$
+
+#math #analysis #series

notes/20241104084927.md

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+# What is asymptotic comparisson of sequences? 
+Asymptotic comparison of two (or more) sequences [[20241002211453]] to one-another is a technique to 
+compare if the sequences act the same when approaching infinity.
+
+#math #analysis #sequence

notes/20241104085206.md

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+# What are the Landau symbols?
+The lambda symbols are used to represent different properties about the relation of two sequences $a_n$ and $b_n$ [[20241002211453]]. [[20241104084927]]
+## big-O
+$a_n = O(b_n)$ means, that there exists a constant C($\in \mathbb{C}$), so that $\forall n \in \mathbb{N}: |\frac{a_n}{b_n}| < C$.
+
+## small-O
+$a_n = o(b_n)$ means that $\lim_{n \to \infty} \frac{a_n}{b_n}=0$ is true.
+
+## Asymptotic equality $\sim$
+$a_n \sim b_n$, means that $\lim_{n \to \infty} \frac{a_n}{b_n}=1$ is true.
+
+#math #analysis #sequence

notes/20241104090341.md

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+# What is combinatorics?
+Combinatorics is a field of math, that is focused on methods to study the structure and size of different (finite) mathematical objects (e.g. sets [[20240929155814]]).
+
+#math #adm #combinatorics