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

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Warm-up-exercises

### Exercise

Compute the first five terms of the Cauchy product of the two convergent series

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\,\,{\text{ and }}\,\,\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}.}$

### Exercise

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.

### Exercise

Let ${\displaystyle {}\sum _{n=0}^{\infty }a_{n}x^{n}}$ and ${\displaystyle {}\sum _{n=0}^{\infty }b_{n}x^{n}}$ be two power series absolutely convergent in ${\displaystyle {}x\in \mathbb {R} }$. Prove that the Cauchy product of these series is exactly

${\displaystyle \sum _{n=0}^{\infty }c_{n}x^{n}{\text{ where }}c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}.}$

### Exercise

Let ${\displaystyle {}x\in \mathbb {R} }$ , ${\displaystyle {}\vert {x}\vert <1}$. Determine (in dependence of ${\displaystyle {}x}$) the sum of the two series

${\displaystyle \sum _{k=0}^{\infty }x^{2k}{\text{ and }}\sum _{k=0}^{\infty }x^{2k+1}.}$

### Exercise

Let

${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}}$

be an absolutely convergent power series. Compute the coefficients of the powers ${\displaystyle {}x^{0},x^{1},x^{2},x^{3},x^{4}}$ in the third power

${\displaystyle \sum _{n=0}^{\infty }c_{n}x^{n}=(\sum _{n=0}^{\infty }a_{n}x^{n})^{3}.}$

### Exercise

Prove that the real function defined by the exponential

${\displaystyle \exp \colon \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto \exp x,}$

has no upper limit and that ${\displaystyle {}0}$ is the infimum (but not the minimum) of the image set.[1]

### Exercise

Prove that for the exponential function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto a^{x},}$

the following calculation rules hold (where ${\displaystyle {}a,b\in \mathbb {R} _{+}}$ and ${\displaystyle {}x,y\in \mathbb {R} }$).

1. ${\displaystyle {}a^{x+y}=a^{x}\cdot a^{y}}$.
2. ${\displaystyle {}a^{-x}={\frac {1}{a^{x}}}}$.
3. ${\displaystyle {}(a^{x})^{y}=a^{xy}}$.
4. ${\displaystyle {}(ab)^{x}=a^{x}b^{x}}$.

### Exercise

Prove that for the logarithm to base ${\displaystyle {}b}$ the following calculation rules hold.

1. We have ${\displaystyle {}\log _{b}(b^{x})=x}$ and ${\displaystyle {}b^{\log _{b}(y)}=y}$, ie, the logarithm to base ${\displaystyle {}b}$ is the inverse to the exponential function to the base ${\displaystyle {}b}$.
2. We have ${\displaystyle {}\log _{b}(y\cdot z)=\log _{b}y+\log _{b}z}$
3. We have ${\displaystyle {}\log _{b}y^{u}=u\cdot \log _{b}y}$ for ${\displaystyle {}u\in \mathbb {R} }$.
4. We have
${\displaystyle {}\log _{a}y=\log _{a}(b^{\log _{b}y})=\log _{b}y\cdot \log _{a}b\,.}$

### Exercise

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

### Exercise

Let ${\displaystyle {}b,c>0}$. Show that

${\displaystyle \operatorname {lim} _{b\rightarrow 0}\,b^{c}=0.}$

Hand-in-exercises

### Exercise (3 points)

Compute the coefficients ${\displaystyle {}c_{0},c_{1},\ldots ,c_{5}}$ of the power series ${\displaystyle {}\sum _{n=0}^{\infty }c_{n}x^{n}}$, which is the Cauchy product of the geometric series with the exponential series.

### Exercise (4 points)

Let

${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}}$

be an absolutely convergent power series. Determine the coefficients of the powers ${\displaystyle {}x^{0},x^{1},x^{2},x^{3},x^{4},x^{5}}$ in the fourth power

${\displaystyle \sum _{n=0}^{\infty }c_{n}x^{n}=(\sum _{n=0}^{\infty }a_{n}x^{n})^{4}.}$

### Exercise (5 points)

For ${\displaystyle {}N\in \mathbb {N} }$ and ${\displaystyle {}x\in \mathbb {R} }$ let

${\displaystyle R_{N+1}(x)=\exp x-\sum _{n=0}^{N}{\frac {x^{n}}{n!}}=\sum _{n=N+1}^{\infty }{\frac {x^{n}}{n!}}}$

be the remainder of the exponential series. Prove that for ${\displaystyle {}\vert {x}\vert \leq 1+{\frac {1}{2}}N}$ the remainder term estimate

${\displaystyle \vert {R_{N+1}(x)}\vert \leq {\frac {2}{(N+1)!}}\vert {x}\vert ^{N+1}}$

holds.

### Exercise (3 points)

Compute by hand the first ${\displaystyle {}4}$ digits in the decimal system of

${\displaystyle \exp 1.}$

### Exercise (4 points)

Prove that the real exponential function defined by the exponential series has the property that for each ${\displaystyle {}d\in \mathbb {N} }$ the sequence

${\displaystyle \left({\frac {\exp n}{n^{d}}}\right)_{n\in \mathbb {N} }}$

diverges to ${\displaystyle {}+\infty }$.[2]

### Exercise (6 points)

Let

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

Fußnoten
1. From the continuity it follows that ${\displaystyle {}\mathbb {R} _{+}}$ is the image set of the real exponential function.
2. Therefore we say that the exponential function grows faster than any polynomial function.