# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 8

*Exercises*

Let be a real convergent sequence and . Show that the sequence is also convergent and

holds.

Let be a real convergent sequence such that for all and . Show that is also convergent and

holds.

Let and be real convergent sequences. Suppose that and for all . Show thatis also convergent and that

holds.

Let . Prove that the sequence converges to .

Let denote the Heron-sequence for the computation of with initial value and let be the Heron-sequence for the computation of with initial value .

- Compute and .
- Compute and .
- Compute and .
- Does the product sequence converge within the rational numbers?

Let . For initial value , let a real sequence be recursively defined by

Show the following statements.

(a) For , we have for all , and the sequence is strictly decreasing.

(b) For , the sequence is constant.

(c) For , we have for all . and the sequence is strictly increasing.

(d) The sequence converges.

(e) The limit is .

Decide whether the sequence

converges in , and determine, if applicable, its limit.

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.

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

converges to .

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

Suppose that for every natural number , a null sequence is given, we denote the -th member of the -th sequence by . Is the sequence , whose -th member is defined by

also a null sequence? Is it possible to apply Lemma 8.1 in this exercise?

Suppose that for every natural number , a null sequence is given, we denote the -th member of the -th sequence by . Is the sequence , whose -th member is defined by

also a null sequence? Is it possible to apply Lemma 8.1 in this exercise?

Discuss the *Cauchy principle of approximation*: If, applying a method of approximations, the approximations do not get much better anymore, thought the effort is increased, then the reason is probably that we are close to the truth. Consider mathematical and and non-mathematical examples and counter-examples.

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

We consider the sequence given by

Show that this is a null sequence.

Show that the sequence with converges.

Let be the sequence of the Fibonacci numbers and

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

Calculate this .

Hint: Show first with the help of the Simpson formula that it is possible to define a sequence of nested intervals with these fractions.

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 real number. We consider the real sequence

(with ).

- Show that the sequence is decreasing.
- Show that all members of the sequence are .
- Show that the sequence converges to .

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

consists of exactly one point .

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.

Give an example of a sequence of closed intervals ()

such that is a null sequence, such that the intersection consists in exactly one point, but such that is not a family of nested intervals.

Show, using Bernoullis's inequality, that the sequence

is increasing.

With a similar argument we can show that the sequence is decreasing and that defines a sequence of nested intervals. The real number defined by these nested intervals is the Euler's number . During this course we will encounter another description for this number.

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

Let be a real number with . Show that the sequence converges to .

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

Examine the convergence of the following sequence

where .

Prove that the sequence diverges to .

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

*Hand-in-exercises*

### Exercise (3 marks)

Determine the limit of the real sequence given by

### Exercise (3 marks)

Determine the limit of the real sequence given by

### Exercise (3 marks)

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

- converges to ,
- converges to ,
- diverges.

### Exercise (3 marks)

Decide whether the sequence

converges and in case determine the limit.

### Exercise (5 marks)

Examine the convergence of the following real sequence .

### Exercise (4 marks)

Let be a convergent sequence with limit . Show that the sequence converges to .

### 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.

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