Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 17/en/latex

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\setcounter{section}{17}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Compute the first five terms of the Cauchy product of the two convergent series
\mathdisp {\sum_{n=1}^\infty { \frac{ 1 }{ n^2 } } \text{ and } \sum_{n=1}^\infty { \frac{ 1 }{ n^3 } }} { . }

}
{} {}




\inputexercise
{}
{

Keep in mind that the partial sums of the Cauchy product of two series are not the product of the partial sums of the two series.

}
{} {}




\inputexercise
{}
{

Let \mathkor {} {\sum _{ n= 0}^\infty a_n x^{ n }} {and} {\sum _{ n= 0}^\infty b_n x^{ n }} {} be two power series absolutely convergent in $x \in \R$. Prove that the Cauchy product of these series is exactly
\mathdisp {\sum _{ n= 0}^\infty c_n x^{ n } \text{ where } c_n = \sum_{i=0}^{n} a_i b_{n-i}} { . }

}
{} {}




\inputexercise
{}
{

Let
\mathbed {x \in \R} {,}
{\betrag { x } <1} {}
{} {} {} {.} Determine \zusatzklammer {in dependence of $x$} {} {} the sum of the two series
\mathdisp {\sum_{k=0 }^\infty x^{2k} \text{ and } \sum_{k=0 }^\infty x^{2k+1}} { . }

}
{} {}




\inputexercise
{}
{

Let
\mathdisp {\sum _{ n= 0}^\infty a_n x^{ n }} { }
be an absolutely convergent power series. Compute the coefficients of the powers
\mathl{x^0,x^1,x^2,x^3,x^4}{} in the third power
\mathdisp {\sum _{ n= 0}^\infty c_n x^{ n } = ( \sum _{ n= 0}^\infty a_n x^{ n } )^3} { . }

}
{} {}




\inputexercise
{}
{

Prove that the real function defined by the exponential \maabbeledisp {\exp} {\R} {\R } {x} { \exp x } {,} has no upper limit and that $0$ is the infimum (but not the minimum) of the image set.\zusatzfussnote {From the continuity it follows that $\R_+$ is the image set of the real exponential function} {.} {}

}
{} {}




\inputexercise
{}
{

Prove that for the exponential function \maabbeledisp {} {\R} {\R } {x} {a^x } {,} the following calculation rules hold \zusatzklammer {where
\mathl{a,b \in \R_+}{} and
\mathl{x,y \in \R}{}} {} {.} \aufzaehlungvier{
\mathl{a^{x+y} = a^x \cdot a^y}{.} }{
\mathl{a^{-x} = { \frac{ 1 }{ a^x } }}{.} }{
\mathl{(a^x)^y = a^{xy}}{.} }{
\mathl{(ab)^x = a^x b^x}{.} }

}
{} {}




\inputexercise
{}
{

Prove that for the logarithm to base $b$ the following calculation rules hold. \aufzaehlungvier{We have \mathkor {} {\log_b(b^x) =x} {and} {b^{\log_b(y)} =y} {,} ie, the logarithm to base $b$ is the inverse to the exponential function to the base $b$. }{We have
\mathl{\log_{ b } (y \cdot z) = \log_{ b } y + \log_{ b } z}{} }{We have
\mathl{\log_{ b } y^u = u \cdot \log_{ b } y}{} for
\mathl{u \in \R}{.} }{We have
\mavergleichskettedisp
{\vergleichskette
{\log_{ a } y }
{ =} { \log_{ a } ( b^{ \log_{ b } y }) }
{ =} {\log_{ b } y \cdot \log_{ a } b }
{ } { }
{ } { }
} {}{}{.} }

}
{} {}




\inputexercise
{}
{

A monetary community has an annual inflation of $2 \%$. After what period of time \zusatzklammer {in years and days} {} {}, the prices have doubled?

}
{} {}




\inputexercise
{}
{

Let
\mathl{b,c >0}{.} Show that
\mathdisp {\operatorname{lim}_{ b \rightarrow 0 } \, b^c =0} { . }

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{3}
{

Compute the coefficients
\mathl{c_0,c_1 , \ldots , c_5}{} of the power series
\mathl{\sum_{n=0}^\infty c_nx^n}{,} which is the Cauchy product of the geometric series with the exponential series.

}
{} {}




\inputexercise
{4}
{

Let
\mathdisp {\sum _{ n= 0}^\infty a_n x^{ n }} { }
be an absolutely convergent power series. Determine the coefficients of the powers $x^0,x^1,x^2,x^3,x^4,x^5$ in the fourth power
\mathdisp {\sum _{ n= 0}^\infty c_n x^{ n } = (\sum _{ n= 0}^\infty a_n x^{ n } )^4} { . }

}
{} {}




\inputexercise
{5}
{

For $N \in \N$ and $x \in \R$ let
\mathdisp {R_{N+1} (x) = \exp x - \sum_{n=0}^N \frac{ x^n}{n!} = \sum_{n=N+1}^\infty \frac{ x^n}{n!}} { }
be the remainder of the exponential series. Prove that for $\betrag { x } \leq 1 + \frac{1}{2}N$ the remainder term estimate
\mathdisp {\betrag { R_{N+1}(x) } \leq \frac{2}{(N+1)!} \betrag { x } ^{N+1}} { }
holds.

}
{} {}




\inputexercise
{3}
{

Compute by hand the first $4$ digits in the decimal system of
\mathdisp {\exp 1} { . }

}
{} {}




\inputexercise
{4}
{

Prove that the real exponential function defined by the exponential series has the property that for each $d \in \N$ the sequence
\mathdisp {\left( \frac{ \exp n }{n^d} \right)_{ n \in \N }} { }
diverges to $+ \infty$\zusatzfussnote {Therefore we say that the exponential function \stichwort {grows faster} {} than any polynomial function} {.} {.}

}
{} {}




\inputexercise
{6}
{

Let \maabbdisp {f} {\R} {\R } {} be a continuous function $\neq 0$, with the property that
\mathdisp {f(x+y)= f(x) \cdot f(y)} { }
for all $x,y \in \R$. Prove that $f$ is an exponential function, i.e. there exists a $b>0$ such that $f(x)=b^x$.

}
{} {}




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