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

Exercises

Exercise

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?

Exercise

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

Exercise

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

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

Let ${}c\in \mathbb {R} _{+}$ be a positive real number and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be the 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

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

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

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

Exercise

Examine the convergence of the following sequence

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

where ${}n\geq 1$ .

Exercise

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

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

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

Exercise

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

Exercise

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

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

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

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) === 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)

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

(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}}\,.

Attention! All operations are to be done within

{}\mathbb {N}

!

Exercise (3 marks)

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)

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

Prove that the real sequence

{\frac {n}{2^{n}}}

converges to

{}0

.
Hint: Find a suitable estimate for

{}2^{n}

using the binomial theorem.

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.