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

__???:__Mathematical induction

Suppose that for every natural number a statement is given. Suppose further that the following conditions are fulfilled.

- is true.
- For all we have: if holds, then also holds.

__???:__Fundamental theorem of arithmetic (existence)

Every natural number , , has a factorization into prime numbers. That means there exists a representation

with prime numbers .

__???:__Irrationality of the square root of 2

There does not exist a rational number such that its square equals . This means that the real number is irrational.

__???:__Theorem of Euclid (prime numbers)

There exist infinitely many prime numbers.

__???:__Euclidean division (polynomial ring)

Let be a field and let be the polynomial ring over . Let be polynomials with . Then there exist unique polynomials such that

__???:__Linear factor and zero of a polynomial

Let be a field and let be the polynomial ring over . Let be a polynomial and . Then is a zero of if and only if is a multiple of the linear polynomial .

__???:__Number of zeroes of a polynomial

Let be a field and let be the polynomial ring over . Let be a polynomial () of degree . Then has at most zeroes.

__???:__Fundamental theorem of algebra

Every nonconstant polynomial over the complex numbers has a zero.

__???:__Interpolation theorem for polynomials

Let be a field, and let different elements and elements are given. Then there exist a unique polynomial of degree , such that holds for all .

__???:__Squeeze criterion

Let and denote real sequences. Suppose that

and that and converge to the same limit . Then also converges to .

__???:__Rules for convergent sequences

Let and be

convergent sequences. Then the following statements hold.- The sequence is convergent, and
holds.

- The sequence is convergent, and
holds.

- For
,
we have
- Suppose that
and
for all
.
Then is also convergent, and
holds.

- Suppose that
and that
for all
.
Then is also convergent, and
holds.

__???:__Real subset bounded from above

Every nonempty subset of the real numbers, which is bounded from above, has a supremum in .

__???:__Cauchy criterion for series

Let

be a
series
of
real numbers. Then the series is
convergent
if and only if the following *Cauchy-criterion* holds: For every
there exists some such that for all

the estimate

holds.

__???:__Behavior of series members in case of convergence

__???:__Leibniz criterion for alternating series

Let be an decreasing null sequence of nonnegative real numbers. Then the series converges.

__???:__Direct comparison test

Let be a convergent series of real numbers and a sequence of real numbers fulfilling for all . Then the series

__???:__Geometric series

For all real numbers with , the geometric series converges absolutely, and the sum equals

__???:__Ratio test

Let

be a series of real numbers. Suppose there exists a real number with , and a with

for all (in particular for ). Then the series converges absolutely.