Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Important theorems/Random

???:Mathematical induction

Suppose that for every natural number ${}n$ a statement ${}A(n)$ is given. Suppose further that the following conditions are fulfilled.

1. ${}A(0)$ is true.
2. For all ${}n$ we have: if ${}A(n)$ holds, then also ${}A(n+1)$ holds.

Then ${}A(n)$ holds for all ${}n$.

???:Fundamental theorem of arithmetic (existence)

Every natural number ${}n\in \mathbb {N}$, ${}n\geq 2$, has a factorization into prime numbers. That means there exists a representation

${}n=p_{1}\cdot p_{2}\cdots p_{r}\,$

with prime numbers ${}p_{i}$.

???:Irrationality of the square root of 2

There does not exist a rational number such that its square equals ${}2$. This means that the real number ${}{\sqrt {2}}$ is irrational.

???:Theorem of Euclid (prime numbers)

There exist infinitely many prime numbers.

???:Binomial theorem

Let ${}a,b$ be elements of a field and let ${}n$ denote a natural number. Then

$(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}$

holds.

???:Algebraic structure of the complex numbers

The complex numbers form a field.

???:Euclidean division (polynomial ring)

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P,T\in K[X]$ be polynomials with ${}T\neq 0$. Then there exist unique polynomials ${}Q,R\in K[X]$ such that

$P=TQ+R{\text{ and with }}\operatorname {deg} \,(R)<\operatorname {deg} \,(T){\text{ or }}R=0.$

???:Linear factor and zero of a polynomial

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P\in K[X]$ be a polynomial and ${}a\in K$. Then ${}a$ is a zero of ${}P$ if and only if ${}P$ is a multiple of the linear polynomial ${}X-a$.

???:Number of zeroes of a polynomial

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}P\in K[X]$ be a polynomial (${}\neq 0$) of degree ${}d$. Then ${}P$ has at most ${}d$ zeroes.

???:Fundamental theorem of algebra

Every nonconstant polynomial ${}P\in \mathbb {C} [X]$ over the complex numbers has a zero.

???:Interpolation theorem for polynomials

Let ${}K$ be a field, and let ${}n$ different elements ${}a_{1},\ldots ,a_{n}\in K$ and ${}n$ elements ${}b_{1},\ldots ,b_{n}\in K$ are given. Then there exist a unique polynomial ${}P\in K[X]$ of degree ${}\leq n-1$, such that ${}P{\left(a_{i}\right)}=b_{i}$ holds for all ${}i$.

???:Uniqueness of limit

A real sequence has at most one limit.

???:Convergent sequence is ...

A convergent real sequence is bounded.

???:Squeeze criterion

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(z_{n}\right)}_{n\in \mathbb {N} }$ denote real sequences. Suppose that

$x_{n}\leq y_{n}\leq z_{n}{\text{ for all }}n\in \mathbb {N}$

and that ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(z_{n}\right)}_{n\in \mathbb {N} }$ converge to the same limit ${}a$. Then also ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ converges to ${}a$.

???:Rules for convergent sequences

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be

convergent sequences. Then the following statements hold.

1. The sequence ${}{\left(x_{n}+y_{n}\right)}_{n\in \mathbb {N} }$ is convergent, and

   ${}\lim _{n\rightarrow \infty }{\left(x_{n}+y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}+{\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,$

   holds.

2. The sequence ${}{\left(x_{n}\cdot y_{n}\right)}_{n\in \mathbb {N} }$ is convergent, and

   ${}\lim _{n\rightarrow \infty }{\left(x_{n}\cdot y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}\cdot {\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,$

   holds.

3. For ${}c\in \mathbb {R}$, we have

   ${}\lim _{n\rightarrow \infty }cx_{n}=c{\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,.$

4. Suppose that ${}\lim _{n\rightarrow \infty }x_{n}=x\neq 0$ and ${}x_{n}\neq 0$ for all ${}n\in \mathbb {N}$. Then ${}\left({\frac {1}{x_{n}}}\right)_{n\in \mathbb {N} }$ is also convergent, and

   ${}\lim _{n\rightarrow \infty }{\frac {1}{x_{n}}}={\frac {1}{x}}\,$

   holds.

5. Suppose that ${}\lim _{n\rightarrow \infty }x_{n}=x\neq 0$ and that ${}x_{n}\neq 0$ for all ${}n\in \mathbb {N}$. Then ${}\left({\frac {y_{n}}{x_{n}}}\right)_{n\in \mathbb {N} }$ is also convergent, and

   ${}\lim _{n\rightarrow \infty }{\frac {y_{n}}{x_{n}}}={\frac {\lim _{n\rightarrow \infty }y_{n}}{x}}\,$

   holds.

???:Real subset bounded from above

Every nonempty subset of the real numbers, which is bounded from above, has a supremum in ${}\mathbb {R}$.

???:Bounded increasing real sequence ...

A bounded and monotone sequence in ${}\mathbb {R}$ converges.

???:Cauchy criterion for series

Let

$\sum _{k=0}^{\infty }a_{k}$

be a series of real numbers. Then the series is convergent if and only if the following Cauchy-criterion holds: For every ${}\epsilon >0$ there exists some ${}n_{0}$ such that for all

${}n\geq m\geq n_{0}\,$

the estimate

${}\vert {\sum _{k=m}^{n}a_{k}}\vert \leq \epsilon \,$

holds.

???:Behavior of series members in case of convergence

Let

$\sum _{k=0}^{\infty }a_{k}$

denote a convergent series of real numbers. Then

${}\lim _{k\rightarrow \infty }a_{k}=0\,.$

???:Leibniz criterion for alternating series

Let ${}{\left(x_{k}\right)}_{k\in \mathbb {N} }$ be an decreasing null sequence of nonnegative real numbers. Then the series ${}\sum _{k=0}^{\infty }(-1)^{k}x_{k}$ converges.

???:Absolute convergence and convergence

An absolutely convergent series of real numbers converges.

???:Direct comparison test

Let ${}\sum _{k=0}^{\infty }b_{k}$ be a convergent series of real numbers and ${}{\left(a_{k}\right)}_{k\in \mathbb {N} }$ a sequence of real numbers fulfilling ${}\vert {a_{k}}\vert \leq b_{k}$ for all ${}k$. Then the series

$\sum _{k=0}^{\infty }a_{k}$

is absolutely convergent.

???:Geometric series

For all real numbers ${}x$ with ${}\vert {x}\vert <1$, the geometric series ${}\sum _{k=0}^{\infty }x^{k}$ converges absolutely, and the sum equals

${}\sum _{k=0}^{\infty }x^{k}={\frac {1}{1-x}}\,.$

???:Ratio test

Let

$\sum _{k=0}^{\infty }a_{k}$

be a series of real numbers. Suppose there exists a real number ${}q$ with ${}0\leq q<1$, and a ${}k_{0}$ with

${}\vert {\frac {a_{k+1}}{a_{k}}}\vert \leq q\,$

for all ${}k\geq k_{0}$ (in particular ${}a_{k}\neq 0$ for ${}k\geq k_{0}$). Then the series ${}\sum _{k=0}^{\infty }a_{k}$ converges absolutely.