# 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

$\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\,\,{\text{ and }}\,\,\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}.$ ### 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 ${}\sum _{n=0}^{\infty }a_{n}x^{n}$ and ${}\sum _{n=0}^{\infty }b_{n}x^{n}$ be two power series absolutely convergent in ${}x\in \mathbb {R}$ . Prove that the Cauchy product of these series is exactly

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

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

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

Let

$\sum _{n=0}^{\infty }a_{n}x^{n}$ be an absolutely convergent power series. Compute the coefficients of the powers ${}x^{0},x^{1},x^{2},x^{3},x^{4}$ in the third power

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

Prove that the real function defined by the exponential

$\exp \colon \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto \exp x,$ has no upper limit and that ${}0$ is the infimum (but not the minimum) of the image set.

### Exercise

Prove that for the exponential function

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

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

### Exercise

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

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

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

### Exercise

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

$\operatorname {lim} _{b\rightarrow 0}\,b^{c}=0.$ Hand-in-exercises

### Exercise (3 points)

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

### Exercise (4 points)

Let

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

$\sum _{n=0}^{\infty }c_{n}x^{n}=(\sum _{n=0}^{\infty }a_{n}x^{n})^{4}.$ ### Exercise (5 points)

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

$R_{N+1}(x)=\exp x-\sum _{n=0}^{N}{\frac {x^{n}}{n!}}=\sum _{n=N+1}^{\infty }{\frac {x^{n}}{n!}}$ be the remainder of the exponential series. Prove that for ${}\vert {x}\vert \leq 1+{\frac {1}{2}}N$ the remainder term estimate

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

### Exercise (3 points)

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

$\exp 1.$ ### Exercise (4 points)

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

$\left({\frac {\exp n}{n^{d}}}\right)_{n\in \mathbb {N} }$ diverges to ${}+\infty$ .

### Exercise (6 points)

Let

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

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