Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 12/en

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Warm-up-exercises

Exercise

Prove that in there is no element such that .


Exercise

Calculate by hand the approximations in the Heron process for the square root of with initial value .


Exercise

Let be a real sequence. Prove that the sequence converges to if and only if for all a natural number exists, such that for all the estimation holds.


Exercise

Examine the convergence of the following sequence

where .


Exercise

Let and be two convergent real sequences with for all . Prove that holds.


Exercise

Let and be three real sequences. Let and and be convergent to the same limit . Prove that also converges to the same limit .


Exercise

Let be a convergent sequence of real numbers with limit equal to . Prove that also the sequence

converges, and specifically to .


The next two exercises concern the Fibonacci numbers.

The sequence of the Fibonacci numbers is defined recursively as


Exercise

Prove by induction the Simpson formula or Simpson identity for the Fibonacci numbers . It says ()


Exercise

Prove by induction the Binet formula for the Fibonacci numbers. This says that

holds ().


Exercise

Examine for each of the following subsets the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.

  1. ,
  2. ,
  3. ,
  4. ,
  5. ,
  6. ,
  7. ,
  8. ,
  9. .




Hand-in-exercises

Exercise (3 points)

Examine the convergence of the following sequence

where .


Exercise (3 points)

Determine the limit of the real sequence given by


Exercise (4 points)

Prove that the real sequence

converges to .


Exercise (6 points)

Examine the convergence of the following real sequence


Exercise (5 points)

Let and be sequences of real numbers and let the sequence be defined as and . Prove that converges if and only if and converge to the same limit.


Exercise (3 points)

Determine the limit of the real sequence given by




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