# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 12

    Warm-up-exercises

### Exercise

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

### Exercise

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

### Exercise

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.

### Exercise

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

### Exercise

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

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

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

### Exercise *

Prove, by induction, the Simpson formula (or Simpson identity) for the Fibonacci numbers $f_{n}$. It says ($n\geq 2$)
    $f_{n+1}f_{n-1}-f_{n}^{2}=(-1)^{n}\,.$

### Exercise

Prove by induction the Binet formula for the Fibonacci numbers. This says that
    $f_{n}={\frac {{\left({\frac {1+{\sqrt {5}}}{2}}\right)}^{n}-{\left({\frac {1-{\sqrt {5}}}{2}}\right)}^{n}}{\sqrt {5}}}\,$
holds ($n\geq 1$).

### Exercise

Examine for each of the following subsets $M\subseteq \mathbb {R}$
the concepts upper bound, lower bound, supremum, infimum, maximum and minimum.
* $\{2,-3,-4,5,6,-1,1\}$,
* $\left\{{\frac {1}{2}},{\frac {-3}{7}},{\frac {-4}{9}},{\frac {5}{9}},{\frac {6}{13}},{\frac {-1}{3}},{\frac {1}{4}}\right\}$,
* $]-5,2]$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}$,
* ${\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\cup \{0\}$,
* $\mathbb {Q} _{-}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 2\right\}}$,
* ${\left\{x\in \mathbb {Q} \mid x^{2}\leq 4\right\}}$,
* ${\left\{x^{2}\mid x\in \mathbb {Z} \right\}}$.
    Hand-in-exercises

### Exercise (3 marks)

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

### Exercise (3 marks)

Determine the limit of the real sequence given by
    $x_{n}={\frac {7n^{3}-3n^{2}+2n-11}{13n^{3}-5n+4}}\,.$

### Exercise (4 marks)

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

### Exercise (5 marks)

Examine the convergence of the following real sequence $x_{n}={\frac {{\sqrt {n}}^{n}}{n!}}$.

### Exercise (5 marks)

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.

### Exercise (3 marks)

Determine the limit of the real sequence given by
    $x_{n}={\frac {2n+5{\sqrt {n}}+7}{-5n+3{\sqrt {n}}-4}}\,.$