# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 7

*Exercises*

We want to find a rational approximation for the number , such that the deviation of the true value should be at most . How good has to be an approximation for such that we obtain the asked for approximation?

Is there any relation between the Din-Norm for paper and square roots?

Compute by hand the approximations in Heron's method to find the square root of with initial value .

Do the first three steps in Heron's method to compute the square root of with initial value (so the approximations for shall be computed; these numbers have to be given as fractions reduced to lowest terms).

Let be a positive real number and let be the Heron-sequence for the computation of with the initial value . Let , , and let be the Heron-sequence for the computation of with initial value . Show that

holds for all .

Determine for the sequence

and

for which (minimal) the estimate

holds.

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.

Negate the statement that a sequence converges in to by transforming the quantifiers.

Examine the convergence of the following sequence

where .

Regarding the sequence , somebody says: "The numerator and the denominator are both going to infinity. However, the denominator is much faster, therefore the sequence converges to “. What do you think about this argument?

Determine whether the following subsets are bounded or not.

- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .

Let be a real number. Show that the sequence is not bounded.

Let be a null sequence and let be a bounded real sequence. Show that also the product sequence is a null sequence.

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 the sequence be given by

- Determine and .
- Does this sequence converge in ?

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

holds ().

*Hand-in-exercises*

===Exercise (3 marks) === Compute by hand the approximations in Heron's method to find the square root of with initial value .

### Exercise (5 marks)

Write a computer-program (pseudocode) for the computation of rational approximations for the square root of a rational number using Heron's method.

- The computer has as many memory units as needed, which can contain natural numbers.

- It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.

- It can add the content of two memory units and write the result into another memory unit.

- It can multiply the content of two memory units and write the result into another memory unit.

- It can print contents of memory units and it can print given texts.

- There is a stop command.

The initial configuration is

with . Here, is the number from which we want to compute the square root, is the initial value and is the wished-for accuracy. The program shall compute and print the Heron-sequence (the numerators and denominators are printed successively) and it shall stop when the member printed last fulfills the property

Attention! All operations are to be done within !

### Exercise (3 marks)

Determine for the sequence

and for

for which (minimal) the estimate

holds.

### Exercise (6 marks)

Let be a convergent real sequence with limit . Show that the sequence defined by

also converges to .

Hint: reduce to the case
.

### Exercise (4 marks)

Prove that the real sequence

Hint: Find a suitable estimate for using the binomial theorem.

### 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.

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