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

Exercises

### Exercise

We want to find a rational approximation for the number ${\displaystyle {}1000000\pi }$, such that the deviation of the true value should be at most ${\displaystyle {}{\frac {1}{1000}}}$. How good has to be an approximation for ${\displaystyle {}\pi }$ such that we obtain the asked for approximation?

### Exercise

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

### Exercise

Compute by hand the approximations ${\displaystyle {}x_{1},x_{2},x_{3},x_{4}}$ in Heron's method to find the square root of ${\displaystyle {}5}$ with initial value ${\displaystyle {}x_{0}=2}$.

### Exercise

Do the first three steps in Heron's method to compute the square root of ${\displaystyle {}b=7}$ with initial value ${\displaystyle {}x_{0}=3}$ (so the approximations ${\displaystyle {}x_{1},x_{2},x_{3}}$ for ${\displaystyle {}{\sqrt {7}}}$ shall be computed; these numbers have to be given as fractions reduced to lowest terms).

### Exercise

Let ${\displaystyle {}c\in \mathbb {R} _{+}}$ be a positive real number and let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be the Heron-sequence for the computation of ${\displaystyle {}{\sqrt {c}}}$ with the initial value ${\displaystyle {}x_{0}\in \mathbb {R} _{+}}$. Let ${\displaystyle {}u\in \mathbb {R} _{+}}$, ${\displaystyle {}d=c\cdot u^{2}}$, ${\displaystyle {}y_{0}=ux_{0}}$ and let ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ be the Heron-sequence for the computation of ${\displaystyle {}{\sqrt {d}}}$ with initial value ${\displaystyle {}y_{0}}$. Show that

${\displaystyle {}y_{n}=ux_{n}\,}$

holds for all ${\displaystyle {}n\in \mathbb {N} }$.

### Exercise

Determine for the sequence

${\displaystyle {}x_{n}:={\frac {2}{3n+5}}\,}$

and

${\displaystyle {}\epsilon ={\frac {1}{10}},\,{\frac {1}{100}},\,{\frac {1}{1000}},\,{\frac {1}{10000}},\ldots \,,}$

for which (minimal) ${\displaystyle {}n}$ the estimate

${\displaystyle {}x_{n}\leq \epsilon \,}$

holds.

### Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be a real sequence. Prove that the sequence converges to ${\displaystyle {}x}$ if and only if for all ${\displaystyle {}k\in \mathbb {N} _{+}}$ a natural number ${\displaystyle {}n_{0}\in \mathbb {N} }$ exists, such that for all ${\displaystyle {}n\geq n_{0}}$ the estimation ${\displaystyle {}\vert {x_{n}-x}\vert \leq {\frac {1}{k}}}$ holds.

### Exercise

Negate the statement that a sequence ${\displaystyle {}x_{n}}$ converges in ${\displaystyle {}\mathbb {R} }$ to ${\displaystyle {}x}$ by transforming the quantifiers.

### Exercise

Examine the convergence of the following sequence

${\displaystyle {}x_{n}={\frac {1}{n^{2}}}\,}$

where ${\displaystyle {}n\geq 1}$.

### Exercise

Regarding the sequence ${\displaystyle {}x_{n}:={\frac {n}{2n}}}$, somebody says: "The numerator and the denominator are both going to infinity. However, the denominator is much faster, therefore the sequence converges to ${\displaystyle {}0}$“. What do you think about this argument?

### Exercise

Determine whether the following subsets ${\displaystyle {}M\subseteq \mathbb {R} }$ are bounded or not.

1. ${\displaystyle {}\mathbb {N} }$,
2. ${\displaystyle {}\left\{{\frac {1}{2}},{\frac {-3}{7}},{\frac {-4}{9}},{\frac {5}{9}},{\frac {6}{13}},{\frac {-1}{3}},{\frac {1}{4}}\right\}}$,
3. ${\displaystyle {}]{-5},2]}$,
4. ${\displaystyle {}{\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}}$,
5. ${\displaystyle {}{\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\cup \{0\}}$,
6. ${\displaystyle {}\mathbb {Q} _{-}}$,
7. ${\displaystyle {}{\left\{x\in \mathbb {Q} \mid x^{2}\leq 2\right\}}}$,
8. ${\displaystyle {}{\left\{x\in \mathbb {Q} \mid x^{2}\geq 4\right\}}}$,
9. ${\displaystyle {}{\left\{x^{2}\mid x\in \mathbb {Z} \right\}}}$.

### Exercise

Let ${\displaystyle {}x>1}$ be a real number. Show that the sequence ${\displaystyle {}x_{n}:=x^{n}}$ is not bounded.

### Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be a null sequence and let ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ be a bounded real sequence. Show that also the product sequence ${\displaystyle {}(x_{n}y_{n})_{n\in \mathbb {N} }}$ is a null sequence.

### Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ be convergent real sequences with ${\displaystyle {}x_{n}\geq y_{n}}$ for all ${\displaystyle {}n\in \mathbb {N} }$. Prove that ${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}}$ holds.

### Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(z_{n}\right)}_{n\in \mathbb {N} }}$ be three real sequences. Let ${\displaystyle {}x_{n}\leq y_{n}\leq z_{n}}$ for all ${\displaystyle {}n\in \mathbb {N} }$ and ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(z_{n}\right)}_{n\in \mathbb {N} }}$ be convergent to the same limit ${\displaystyle {}a}$. Prove that also ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ converges to the same limit ${\displaystyle {}a}$.

### Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be a convergent sequence of real numbers with limit equal to ${\displaystyle {}x}$. Prove that also the sequence
${\displaystyle {\left(\vert {x_{n}}\vert \right)}_{n\in \mathbb {N} }}$
converges, and specifically to ${\displaystyle {}\vert {x}\vert }$.

### Exercise

Let the sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be given by

${\displaystyle {}x_{n}={\begin{cases}1,\,{\text{ falls }}n{\text{ is a prime number }}\,,\\0\,{\text{ else}}\,.\end{cases}}\,}$
1. Determine ${\displaystyle {}x_{117}}$ and ${\displaystyle {}x_{127}}$.
2. Does this sequence converge in ${\displaystyle {}\mathbb {Q} }$?

### Exercise

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

${\displaystyle {}f_{n}={\frac {{\left({\frac {1+{\sqrt {5}}}{2}}\right)}^{n}-{\left({\frac {1-{\sqrt {5}}}{2}}\right)}^{n}}{\sqrt {5}}}\,}$

holds (${\displaystyle {}n\geq 1}$).

Hand-in-exercises

===Exercise (3 marks) === Compute by hand the approximations ${\displaystyle {}x_{1},x_{2},x_{3},x_{4}}$ in Heron's method to find the square root of ${\displaystyle {}7}$ with initial value ${\displaystyle {}x_{0}=2}$.

### 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

${\displaystyle (a,b,c,d,e,0,0,\ldots )}$

with ${\displaystyle {}b,c,e\neq 0}$. Here, ${\displaystyle {}a/b}$ is the number from which we want to compute the square root, ${\displaystyle {}x_{0}=c}$ is the initial value and ${\displaystyle {}d/e}$ is the wished-for accuracy. The program shall compute and print the Heron-sequence ${\displaystyle {}x_{0},x_{1},x_{2},\ldots }$ (the numerators and denominators are printed successively) and it shall stop when the member ${\displaystyle {}x_{n}}$ printed last fulfills the property

${\displaystyle {}\vert {x_{n}^{2}-{\frac {a}{b}}}\vert \leq {\frac {d}{e}}\,.}$

Attention! All operations are to be done within ${\displaystyle {}\mathbb {N} }$!

### Exercise (3 marks)

Determine for the sequence

${\displaystyle {}x_{n}:={\frac {2n+1}{3n-4}}\,}$

and for

${\displaystyle {}\epsilon ={\frac {1}{10}},\,{\frac {1}{100}},\,{\frac {1}{1000}}\,,}$

for which (minimal) ${\displaystyle {}n}$ the estimate

${\displaystyle {}\vert {x_{n}-{\frac {2}{3}}}\vert \leq \epsilon \,}$

holds.

### Exercise (6 marks)

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be a convergent real sequence with limit ${\displaystyle {}x}$. Show that the sequence defined by

${\displaystyle {}y_{n}:={\frac {x_{0}+x_{1}+\cdots +x_{n}}{n+1}}\,}$

also converges to ${\displaystyle {}x}$.
Hint: reduce to the case ${\displaystyle {}x=0}$.

### Exercise (4 marks)

Prove that the real sequence

${\displaystyle {\frac {n}{2^{n}}}}$
converges to ${\displaystyle {}0}$.
Hint: Find a suitable estimate for ${\displaystyle {}2^{n}}$ using the binomial theorem.

### Exercise (5 marks)

Let ${\displaystyle {}(x_{n})_{n\in \mathbb {N} }}$ and ${\displaystyle {}(y_{n})_{n\in \mathbb {N} }}$ be sequences of real numbers and let the sequence ${\displaystyle {}(z_{n})_{n\in \mathbb {N} }}$ be defined as ${\displaystyle {}z_{2n-1}:=x_{n}}$ and ${\displaystyle {}z_{2n}:=y_{n}}$. Prove that ${\displaystyle {}(z_{n})_{n\in \mathbb {N} }}$ converges if and only if ${\displaystyle {}(x_{n})_{n\in \mathbb {N} }}$ and ${\displaystyle {}(y_{n})_{n\in \mathbb {N} }}$ converge to the same limit.