Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 13



Warm-up-exercises

Prove the statements (1), (3) and (5) of the rules.


Let  . Prove that the sequence   converges to  .


Give an example of a Cauchy sequence in  , such that (in  ) it does not converge.


Give an example of a real sequence, that does not converge, but it contains a convergent subsequence.


Let   be a non-negative real number and  . Prove that the sequence defined recursively as   and

 

converges to  .


Let   be the sequence of the Fibonacci numbers and

 

Prove that this sequence converges in   and that its limit   satisfies the relation

 

Calculate this  .


Let   and   be two non-negative real numbers. Prove that the arithmetic mean of these numbers is larger than or equal to their geometric mean.


Let  ,  , be a sequence of nested intervals in  . Prove that the intersection

 

consists of exactly one point  .


Let   be a real number. Prove that the sequence  , diverges to  .


Give an example of a real sequence  , such that it contains a subsequence that diverges to   and also a subsequence that diverges to  .




Hand-in-exercises

Exercise (3 marks)

Give examples of convergent sequences of real numbers   and   with  ,  , and with   such that the sequence

 
  1. converges to  ,
  2. converges to  ,
  3. diverges.


Exercise (5 marks)

Let   and   be polynomials with  . Determine, depending on   and  , whether

 

(which is defined for   sufficiently large) is a convergent sequence or not, and determine the limit in the convergent case.


Exercise (4 marks)

Let  ,  , be a sequence of nested intervals in   and let   be a real sequence with   for all  . Prove that this sequence converges to the unique number belonging to the intersection of the family of nested intervals.


Exercise (4 marks)

Let   be positive real numbers. We define recursively two sequences   and   such that  ,  , and that

 
 

Prove that   is a sequence of nested intervals.


Exercise (2 marks)

Prove that the sequence   diverges to  .


Exercise (3 marks)

Let   be a real sequence with   for all  . Prove that the sequence diverges to   if and only if the sequence   converges to  .


Exercise (10 marks)

There are   persons in a room and they would like to play secret Santa. This means that for each person  , another person   has to be determined to whom   has to give a gift. Every person is only allowed to know who to give the gift, no one is allowed to know more. The people stay the whole time in the room, they do not look away. They only have paper and pens. They are allowed to shuffle and to look in secret at choosen cards. Describe a procedure to determine a gift relation which satisfies all conditions.