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.

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

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

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