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

???:Mathematical induction

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

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

???:Fundamental theorem of arithmetic (existence)

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

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

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

???:Irrationality of the square root of 2

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

???:Theorem of Euclid (prime numbers)

There exist infinitely many prime numbers.

???:Binomial theorem

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

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

holds.

???:Algebraic structure of the complex numbers

The complex numbers form a field.

???:Euclidean division (polynomial ring)

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

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

???:Linear factor and zero of a polynomial

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

???:Number of zeroes of a polynomial

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

???:Fundamental theorem of algebra

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

???:Interpolation theorem for polynomials

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

???:Uniqueness of limit

A real sequence has at most one limit.

???:Convergent sequence is ...

???:Squeeze criterion

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

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

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

???:Rules for convergent sequences

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

convergent sequences. Then the following statements hold.
1. The sequence ${\displaystyle {}{\left(x_{n}+y_{n}\right)}_{n\in \mathbb {N} }}$ is convergent, and
${\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}+y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}+{\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}$

holds.

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

holds.

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

holds.

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

holds.

???:Real subset bounded from above

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

???:Bounded increasing real sequence ...

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

???:Cauchy criterion for series

Let

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

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

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

the estimate

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

holds.

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

Let

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

denote a convergent series of real numbers. Then

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

???:Leibniz criterion for alternating series

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

???:Absolute convergence and convergence

???:Direct comparison test

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

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

???:Geometric series

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

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

???:Ratio test

Let

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

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

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

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