Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 12



Warm-up-exercises

Prove that in   there is no element   such that  .


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


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.


Examine the convergence of the following sequence

 

where  .


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


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


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

 

converges, and specifically to  .


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

 


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

 

holds ( ).


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 marks)

Examine the convergence of the following sequence

 

where  .


Exercise (3 marks)

Determine the limit of the real sequence given by

 


Exercise (4 marks)

Prove that the real sequence

 

converges to  .


Exercise (5 marks)

Examine the convergence of the following real sequence  .


Exercise (5 marks)

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 marks)

Determine the limit of the real sequence given by