Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 12/en/latex

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\zwischenueberschrift{Warm-up-exercises}

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

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\inputexercise
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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 $x_0=2$.

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\inputexercise
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Let
\mathl{{ \left( x_n \right) }_{n \in \N }}{} be a real sequence. Prove that the sequence converges to $x$ if and only if for all
\mathl{k \in \N_+}{} a natural number
\mathl{n_0 \in \N}{} exists, such that for all
\mathl{n \geq n_0}{} the estimation
\mathl{\betrag { x_n-x } \leq { \frac{ 1 }{ k } }}{} holds.

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\inputexercise
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Examine the convergence of the following sequence
\mathdisp {x_n = { \frac{ 1 }{ n^2 } }} { }
where
\mathl{n \geq 1}{.}

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\inputexercise
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Let \mathkor {} {{ \left( x_n \right) }_{n \in \N }} {and} {{ \left( y_n \right) }_{n \in \N }} {} be two convergent real sequences with
\mathl{x_n \geq y_n}{} for all
\mathl{n \in \N}{.} Prove that $\lim_{n \rightarrow \infty} x_n \geq \lim_{n \rightarrow \infty} y_n$ holds.

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\inputexercise
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Let \mathkor {} {{ \left( x_n \right) }_{n \in \N }, \, { \left( y_n \right) }_{n \in \N }} {and} {{ \left( z_n \right) }_{n \in \N }} {} be three real sequences. Let $x_n \leq y_n \leq z_n \text{ for all } n \in \N$ and \mathkor {} {{ \left( x_n \right) }_{n \in \N }} {and} {{ \left( z_n \right) }_{n \in \N }} {} be convergent to the same limit $a$. Prove that also ${ \left( y_n \right) }_{n \in \N }$ converges to the same limit $a$.

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

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The next two exercises concern the Fibonacci numbers.

The sequence of the \stichwort {Fibonacci numbers} {} $f_n$ is defined recursively as
\mathdisp {f_1:=1 \, , f_2:=1 \text{ and } f_{n+2}:=f_{n+1} +f_{n}} { . }

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

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\inputexercise
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Prove by induction the \stichwort {Binet formula} {} for the Fibonacci numbers. This says that
\mathdisp {f_n = \frac{ (\frac{1+\sqrt{5} }{2})^n - (\frac{1-\sqrt{5} }{2})^n}{\sqrt{5} }} { }
holds \zusatzklammer {\mathlk{n \geq 1}{}} {} {.}

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

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\zwischenueberschrift{Hand-in-exercises}

\inputexercise
{3}
{

Examine the convergence of the following sequence
\mathdisp {x_n = { \frac{ 1 }{ \sqrt{n} } }} { , }
where \mathlk{n \geq 1}{.}

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\inputexercise
{3}
{

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

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\inputexercise
{4}
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Prove that the real sequence
\mathdisp {\left( \frac{n}{2^n} \right)_{ n \in \N }} { }
converges to $0$.

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\inputexercise
{6}
{

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

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\inputexercise
{5}
{

Let $(x_n)_{n \in \N}$ and $(y_n)_{n \in \N}$ be sequences of real numbers and let the sequence $(z_n)_{n \in \N}$ be defined as $z_{2n-1}:=x_n$ and $z_{2n}:=y_n$. Prove that $(z_n)_{n \in \N}$ converges if and only if $(x_n)_{n \in \N}$ and $(y_n)_{n \in \N}$ converge to the same limit.

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\inputexercise
{3}
{

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

}
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