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

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???:Mathematical induction

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

  1. is true.
  2. For all we have: if holds, then also holds.
Then holds for all .

???: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.


???:Binomial theorem

Let be elements of a field and let denote a natural number. Then

holds.


???:Algebraic structure of the complex numbers

The complex numbers form a field.


???: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 be given. Then there exists a unique polynomial of degree , such that holds for all .


???:Uniqueness of limit

A real sequence has at most one limit.


???:Convergent sequence is ...


???: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.
  1. The sequence is convergent, and

    holds.

  2. The sequence is convergent, and

    holds.

  3. For , we have
  4. Suppose that and for all . Then is also convergent, and

    holds.

  5. 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 .


???:Bounded increasing real sequence ...

A bounded and monotone sequence in converges.


???: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

Let

denote a convergent series of real numbers. Then


???:Leibniz criterion for alternating series

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


???:Absolute convergence and convergence


???:Direct comparison test

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

is absolutely convergent.


???: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.