# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 12/en

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Warm-up-exercises

### Exercise

Prove that in ${\displaystyle {}\mathbb {Q} }$ there is no element ${\displaystyle {}x}$ such that ${\displaystyle {}x^{2}=2}$.

### Exercise

Calculate by hand the approximations ${\displaystyle {}x_{1},x_{2},x_{3},x_{4}}$ in the Heron process for the square root of ${\displaystyle {}5}$ with initial value ${\displaystyle {}x_{0}=2}$.

### 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

Examine the convergence of the following sequence

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

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

### Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ be two 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 ${\displaystyle {}x_{n}\leq y_{n}\leq z_{n}{\text{ for all }}n\in \mathbb {N} }$ 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 ${\displaystyle {}\vert {x}\vert }$.

The next two exercises concern the Fibonacci numbers.

The sequence of the Fibonacci numbers ${\displaystyle {}f_{n}}$ is defined recursively as

${\displaystyle f_{1}:=1\,,f_{2}:=1{\text{ and }}f_{n+2}:=f_{n+1}+f_{n}.}$

### Exercise

Prove by induction the Simpson formula or Simpson identity for the Fibonacci numbers ${\displaystyle {}f_{n}}$. It says (${\displaystyle n\geq 2}$)

${\displaystyle f_{n+1}f_{n-1}-f_{n}^{2}=(-1)^{n}.}$

### Exercise

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

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

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

### Exercise

Examine for each of the following subsets ${\displaystyle {}M\subseteq \mathbb {R} }$ the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.

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

Hand-in-exercises

### Exercise (3 points)

Examine the convergence of the following sequence

${\displaystyle x_{n}={\frac {1}{\sqrt {n}}},}$

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

### Exercise (3 points)

Determine the limit of the real sequence given by

${\displaystyle x_{n}={\frac {7n^{3}-3n^{2}+2n-11}{13n^{3}-5n+4}}.}$

### Exercise (4 points)

Prove that the real sequence

${\displaystyle \left({\frac {n}{2^{n}}}\right)_{n\in \mathbb {N} }}$

converges to ${\displaystyle {}0}$.

### Exercise (6 points)

Examine the convergence of the following real sequence

${\displaystyle x_{n}={\frac {{\sqrt {n}}^{n}}{n!}}.}$

### Exercise (5 points)

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.

### Exercise (3 points)

Determine the limit of the real sequence given by

${\displaystyle x_{n}={\frac {2n+5{\sqrt {n}}+7}{-5n+3{\sqrt {n}}-4}}.}$