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

    Exercises

### Exercise Create referencenumber

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

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

### Exercise Create referencenumber

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

### Exercise Create referencenumber

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

### Exercise Create referencenumber

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

### Exercise Create referencenumber

Determine for the sequence
    $x_{n}:={\frac {2}{3n+5}}\,$
and
    $\epsilon ={\frac {1}{10}},\,{\frac {1}{100}},\,{\frac {1}{1000}},\,{\frac {1}{10000}},\ldots \,,$
for which (minimal) $n$ the estimate
    $x_{n}\leq \epsilon \,$
holds.
===Exercise Exercise 7.7
Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence. Prove that the sequence converges to $x$ if and only if for all $k\in \mathbb {N} _{+}$ a natural number $n_{0}\in \mathbb {N}$ exists, such that for all $n\geq n_{0}$ the estimation $\vert {x_{n}-x}\vert \leq {\frac {1}{k}}$ holds.

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

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Examine the convergence of the following sequence
    $x_{n}={\frac {1}{n^{2}}}\,$
where $n\geq 1$.

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

### Exercise Create referencenumber

Determine whether the following subsets $M\subseteq \mathbb {R}$ are boundedMDLD/bounded (R) or not.
* $\mathbb {N}$,
* $\left\{{\frac {1}{2}},{\frac {-3}{7}},{\frac {-4}{9}},{\frac {5}{9}},{\frac {6}{13}},{\frac {-1}{3}},{\frac {1}{4}}\right\}$,
* $]{-5},2]$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\cup \{0\}$,
* $\mathbb {Q} _{-}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 2\right\}}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\geq 4\right\}}$,
* ${\left\{x^{2}\mid x\in \mathbb {Z} \right\}}$.

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Let $x>1$ be a
real number.MDLD/real number Show that the sequence $x_{n}:=x^{n}$ is not bounded.MDLD/bounded (R)

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Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a null sequence and let ${\left(y_{n}\right)}_{n\in \mathbb {N} }$ be a bounded real sequence. Show that also the product sequence $(x_{n}y_{n})_{n\in \mathbb {N} }$ is a null sequence.
===Exercise Exercise 7.16
Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(y_{n}\right)}_{n\in \mathbb {N} }$ be convergent real sequences with $x_{n}\geq y_{n}$ for all $n\in \mathbb {N}$. Prove that $\lim _{n\rightarrow \infty }x_{n}\geq \lim _{n\rightarrow \infty }y_{n}$ holds.
===Exercise Exercise 7.17
Let ${\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(z_{n}\right)}_{n\in \mathbb {N} }$ be three real sequences. Let $x_{n}\leq y_{n}\leq z_{n}$ for all $n\in \mathbb {N}$ and ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${\left(z_{n}\right)}_{n\in \mathbb {N} }$ be convergent to the same limit $a$. Prove that also ${\left(y_{n}\right)}_{n\in \mathbb {N} }$ converges to the same limit $a$.

### Exercise Create referencenumber

${\left(x_{n}\right)}_{n\in \mathbb {N} }$
$x$
    ${\left(\vert {x_{n}}\vert \right)}_{n\in \mathbb {N} }$
$\vert {x}\vert$

### Exercise Create referencenumber

Let the sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be given by
    $x_{n}={\begin{cases}1,\,{\text{ falls }}n{\text{ is a prime number }}\,,\\0\,{\text{ else}}\,.\end{cases}}\,$
* Determine $x_{117}$ and $x_{127}$.
* Does this sequence converge in $\mathbb {Q}$?

### Exercise Create referencenumber

Prove by induction the Binet formula for the Fibonacci numbers. This says that
    $f_{n}={\frac {{\left({\frac {1+{\sqrt {5}}}{2}}\right)}^{n}-{\left({\frac {1-{\sqrt {5}}}{2}}\right)}^{n}}{\sqrt {5}}}\,$
holds ($n\geq 1$).
    Hand-in-exercises
===Exercise (3 marks)
Create referencenumber=== Compute by hand the approximations $x_{1},x_{2},x_{3},x_{4}$ in Heron's method to find the square root of $7$ with initial value $x_{0}=2$.

### Exercise (5 marks) Create referencenumber

Write a computer-program (pseudocode) for the computation of rational approximations for the square root of a rational number using Heron's method.MDLD/Heron's method
The initial configuration is
    $(a,b,c,d,e,0,0,\ldots )$
with $b,c,e\neq 0$. Here, $a/b$ is the number from which we want to compute the square root, $x_{0}=c$ is the initial value and $d/e$ is the wished-for accuracy. The program shall compute and print the Heron-sequence $x_{0},x_{1},x_{2},\ldots$ (the numerators and denominators are printed successively) and it shall stop when the member $x_{n}$ printed last fulfills the property
    $\vert {x_{n}^{2}-{\frac {a}{b}}}\vert \leq {\frac {d}{e}}\,.$
$\mathbb {N}$

### Exercise (3 marks) Create referencenumber

Determine for the sequence
    $x_{n}:={\frac {2n+1}{3n-4}}\,$
and for
    $\epsilon ={\frac {1}{10}},\,{\frac {1}{100}},\,{\frac {1}{1000}}\,,$
for which (minimal) $n$ the estimate
    $\vert {x_{n}-{\frac {2}{3}}}\vert \leq \epsilon \,$
holds.

### Exercise (6 marks) Create referencenumber

Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a convergentMDLD/convergent (R) real sequenceMDLD/real sequence with limitMDLD/limit (real sequence) $x$. Show that the sequence defined by
    $y_{n}:={\frac {x_{0}+x_{1}+\cdots +x_{n}}{n+1}}\,$
also converges to $x$. Hint: reduce to the case $x=0$.

### Exercise (4 marks) Create referencenumber

Prove that the real sequence
    ${\frac {n}{2^{n}}}$
$0$
$2^{n}$

### Exercise (5 marks) Create referencenumber

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