Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 8

Exercises

Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real convergent sequence and ${}c\in \mathbb {R}$. Show that the sequence ${}{\left(c\cdot x_{n}\right)}_{n\in \mathbb {N} }$ is also convergent and

${}\lim _{n\rightarrow \infty }{\left(c\cdot x_{n}\right)}=c\cdot {\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,$

holds.

Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real convergent sequence such that ${}x_{n}\neq 0$ for all ${}n\in \mathbb {N}$ and ${}\lim _{n\rightarrow \infty }x_{n}=x\neq 0$. Show that ${}{\left({\frac {1}{x_{n}}}\right)}_{n\in \mathbb {N} }$is also convergent and

${}\lim _{n\rightarrow \infty }{\frac {1}{x_{n}}}={\frac {1}{x}}\,$

holds.

Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be real convergent sequences. Suppose that ${}\lim _{n\rightarrow \infty }x_{n}=x\neq 0$ and ${}x_{n}\neq 0$ for all ${}n\in \mathbb {N}$. Show that${}{\left({\frac {y_{n}}{x_{n}}}\right)}_{n\in \mathbb {N} }$is also convergent and that

${}\lim _{n\rightarrow \infty }{\frac {y_{n}}{x_{n}}}={\frac {\lim _{n\rightarrow \infty }y_{n}}{x}}\,$

holds.

Exercise

Let ${}k\in \mathbb {N} _{+}$. Prove that the sequence ${}{\left({\frac {1}{n^{k}}}\right)}_{n\in \mathbb {N} }$ converges to ${}0$.

Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote the Heron-sequence for the computation of ${}{\sqrt {3}}$ with initial value ${}x_{0}=1$ and let ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be the Heron-sequence for the computation of ${}{\sqrt {\frac {1}{3}}}$ with initial value ${}y_{0}=1$.

1. Compute ${}x_{1}$ and ${}x_{2}$.
2. Compute ${}y_{1}$ and ${}y_{2}$.
3. Compute ${}x_{0}\cdot y_{0},\,x_{1}\cdot y_{1}$ and ${}x_{2}\cdot y_{2}$.
4. Does the product sequence ${}z_{n}=x_{n}\cdot y_{n}$ converge within the rational numbers?

Exercise

Let ${}a\in \mathbb {R}$. For initial value ${}x_{0}\in \mathbb {R}$, let a real sequence be recursively defined by

${}x_{n+1}={\frac {x_{n}+a}{2}}\,.$

Show the following statements.

(a) For ${}x_{0}>a$, we have ${}x_{n}>a$ for all ${}n\in \mathbb {N}$, and the sequence is strictly decreasing.

(b) For ${}x_{0}=a$, the sequence is constant.

(c) For ${}x_{0}<a$, we have ${}x_{n}<a$ for all ${}n\in \mathbb {N}$. and the sequence is strictly increasing.

(d) The sequence converges.

(e) The limit is ${}a$.

Exercise

Decide whether the sequence

${}x_{n}={\frac {6n^{3}+3n^{2}-4n+5}{7n^{3}-6n^{2}-2}}\,$

converges in ${}\mathbb {Q}$, and determine, if applicable, its limit.

Exercise

Let ${}P=\sum _{i=0}^{d}a_{i}x^{i}$ and ${}Q=\sum _{i=0}^{e}b_{i}x^{i}$ be polynomials with ${}a_{d},b_{e}\neq 0$. Determine, depending on ${}d$ and ${}e$, whether

${}z_{n}={\frac {P(n)}{Q(n)}}\,$

(which is defined for ${}n$ sufficiently large) is a convergent sequence or not, and determine the limit in the convergent case.

Exercise

Let ${}a\in \mathbb {R}$ be a non-negative real number and ${}c\in \mathbb {R} _{+}$. Prove that the sequence defined recursively as ${}x_{0}=c$ and

${}x_{n+1}:={\frac {x_{n}+a/x_{n}}{2}}\,,$

converges to ${}{\sqrt {a}}$.

Exercise

Give an example of a real sequence, that does not converge, but it contains a convergent subsequence.

Exercise

Suppose that for every natural number ${}k$, a null sequence ${}y_{k}$ is given, we denote the ${}n$-th member of the ${}k$-th sequence by ${}y_{kn}$. Is the sequence ${}z_{n}$, whose ${}n$-th member is defined by

${}z_{n}=\sum _{k=1}^{n}y_{kn}\,,$

also a null sequence? Is it possible to apply Lemma 8.1   (1) in this exercise?

Exercise

Suppose that for every natural number ${}k$, a null sequence ${}y_{k}$ is given, we denote the ${}n$-th member of the ${}k$-th sequence by ${}y_{kn}$. Is the sequence ${}z_{n}$, whose ${}n$-th member is defined by

${}z_{n}=\prod _{k=1}^{n}y_{kn}\,,$

also a null sequence? Is it possible to apply Lemma 8.1   (3) in this exercise?

Exercise

Discuss the Cauchy principle of approximation: If, applying a method of approximations, the approximations do not get much better anymore, thought the effort is increased, then the reason is probably that we are close to the truth. Consider mathematical and and non-mathematical examples and counter-examples.

Exercise

Give an example of a Cauchy sequence in ${}\mathbb {Q}$, such that (in ${}\mathbb {Q}$) it does not converge.

Exercise

We consider the sequence given by

${}x_{n}={\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{6}}\cdots {\frac {2n-1}{2n}}\,.$

Show that this is a null sequence.

Exercise

Show that the sequence ${}(a_{n})_{n\in \mathbb {N} }$ with ${}a_{n}={\frac {1}{n+1}}+\cdots +{\frac {1}{2n}}$ converges.

Exercise

Let ${}{\left(f_{n}\right)}_{n\in \mathbb {N} }$ be the sequence of the Fibonacci numbers and

${}x_{n}:={\frac {f_{n}}{f_{n-1}}}\,.$

Prove that this sequence converges in ${}\mathbb {R}$ and that its limit ${}x$ satisfies the relation

${}x=1+x^{-1}\,.$

Calculate this ${}x$.
Hint: Show first with the help of the Simpson formula that it is possible to define a sequence of nested intervals with these fractions.

For two nonnegative real numbers ${}x$ and ${}y$, we call

${\sqrt {x\cdot y}}$

their geometric mean.

Exercise *

Let ${}x$ and ${}y$ be two non-negative real numbers. Prove that the arithmetic mean of these numbers is larger than or equal to their geometric mean.

Exercise

Let ${}b\geq 1$ be a real number. We consider the real sequence

${}x_{n}:=b^{\frac {1}{n}}={\sqrt[{n}]{b}}\,$

(with ${}n\in \mathbb {N} _{+}$).

1. Show that the sequence is decreasing.
2. Show that all members of the sequence are ${}\geq 1$.
3. Show that the sequence converges to ${}1$.

Exercise

Let ${}I_{n}$, ${}n\in \mathbb {N}$, be a sequence of nested intervals in ${}\mathbb {R}$. Prove that the intersection

$\bigcap _{n\in \mathbb {N} }I_{n}$

consists of exactly one point ${}x\in \mathbb {R}$.

Exercise

Let ${}I_{n}$, ${}n\in \mathbb {N}$, be a sequence of nested intervals in ${}\mathbb {R}$ and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence with ${}x_{n}\in I_{n}$ for all ${}n\in \mathbb {N}$. Prove that this sequence converges to the unique number belonging to the intersection of the family of nested intervals.

Exercise

Give an example of a sequence of closed intervals (${}n\in \mathbb {N} _{+}$)

${}I_{n}=[a_{n},b_{n}]\subseteq \mathbb {R} \,$

such that ${}b_{n}-a_{n}$ is a null sequence, such that the intersection ${}\bigcap _{n\in \mathbb {N} _{+}}I_{n}$ consists in exactly one point, but such that ${}I_{n}$ is not a family of nested intervals.

Exercise

Show, using Bernoullis's inequality, that the sequence

${}x_{n}={\left(1+{\frac {1}{n}}\right)}^{n}\,$

is increasing.

With a similar argument we can show that the sequence ${}{\left(1+{\frac {1}{n}}\right)}^{n+1}$ is decreasing and that ${}[{\left(1+{\frac {1}{n}}\right)}^{n},{\left(1+{\frac {1}{n}}\right)}^{n+1}]$ defines a sequence of nested intervals. The real number defined by these nested intervals is the Euler's number ${}e$. During this course we will encounter another description for this number.

Exercise

Let ${}x>1$ be a real number. Prove that the sequence ${}x^{n},\,n\in \mathbb {N}$, diverges to ${}+\infty$.

Exercise

Let ${}x$ be a real number with ${}\vert {x}\vert <1$. Show that the sequence ${}x_{n}:=x^{n}$ converges to ${}0$.

Exercise

Give an example of a real sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$, such that it contains a subsequence that diverges to ${}+\infty$ and also a subsequence that diverges to ${}-\infty$.

Exercise

Examine the convergence of the following sequence

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

where ${}n\geq 1$.

Exercise

Prove that the sequence ${}{\left({\sqrt {n}}\right)}_{n\in \mathbb {N} }$ diverges to ${}\infty$.

Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence with ${}x_{n}>0$ for all ${}n\in \mathbb {N}$. Prove that the sequence diverges to ${}+\infty$ if and only if the sequence ${}{\left({\frac {1}{x_{n}}}\right)}_{n\in \mathbb {N} }$ converges to ${}0$.

Hand-in-exercises

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 (3 marks)

Determine the limit of the real sequence given by

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

Exercise (3 marks)

Give examples of convergent sequences of real numbers ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ with ${}x_{n}\neq 0$, ${}n\in \mathbb {N}$, and with ${}\lim _{n\rightarrow \infty }x_{n}=0$ such that the sequence

${\left({\frac {y_{n}}{x_{n}}}\right)}_{n\in \mathbb {N} }$

1. converges to ${}0$,
2. converges to ${}1$,
3. diverges.

Exercise (3 marks)

Decide whether the sequence

${}a_{n}={\sqrt {n+1}}-{\sqrt {n}}\,$

converges and in case determine the limit.

Exercise (5 marks)

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

Exercise (4 marks)

Let ${}x_{n}\in \mathbb {R} _{\geq 0}$ be a convergent sequence with limit ${}x$. Show that the sequence ${}{\sqrt {x_{n}}}$ converges to ${}{\sqrt {x}}$.

Exercise (4 marks)

Let ${}b>a>0$ be positive real numbers. We define recursively two sequences ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ and ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ such that ${}x_{0}=a$, ${}y_{0}=b$, and that

${}x_{n+1}={\text{ geometric mean of }}x_{n}{\text{ and }}y_{n}\,,$

${}y_{n+1}={\text{ arithmetic mean of }}x_{n}{\text{ and }}y_{n}\,.$

Prove that ${}[x_{n},y_{n}]$ is a sequence of nested intervals.

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