Exercises: sequences – Serlo

Aus Wikibooks
Zur Navigation springen Zur Suche springen

Questions: Sequences[Bearbeiten]

Exercise

Which of the following statements are true?

  1. Every index can uniquely be matched to a sequence element .
  2. Every sequence element can uniquely be matched to an index .
  3. A sequence being neither bounded from above nor from below does not exist.
  4. A sequence increasing and decreasing monotonically does not exist.
  5. A sequence increasing and decreasing strictly monotonically does not exist.
  6. A sequence increasing monotonically and being bounded from above does not exist.

Solution

  1. True.
  2. False. A constant sequence is a great counterexample. As all sequence elements , , have the same value, you can match this value to every index and not only one unique index.
  3. False. The sequence is neither bounded from above nor from below.
  4. False. Every constant sequence is monotonically increasing and decreasing. Only „strict monotonicity“ implies that the sequence elements need to be unequal.
  5. True.
  6. False. The sequence with for all is bounded from above, as holds for all . Furthermore, it's monotonically increasing. For all , we have and therefore .

Exercise: Find a sequence[Bearbeiten]

Exercise

Find a sequence fulfilling the following conditions:

  1. for all
  2. for all odd

Write down an explicit and a recursive formula for your sequence!

Solution

One possible sequence is

An explicit formula of this sequence is given by

A recursive formula of this sequence is given by and for all :