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.

1. ${}\mathbb {N}$,
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. ${}{\left\{x\in \mathbb {Q} \mid x^{2}\geq 4\right\}}$,
9. ${}{\left\{x^{2}\mid x\in \mathbb {Z} \right\}}$.

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

${}x_{n}={\begin{cases}1,\,{\text{ falls }}n{\text{ is a prime number }}\,,\\0\,{\text{ else}}\,.\end{cases}}\,$

1. Determine ${}x_{117}$ and ${}x_{127}$.
2. 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.

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