# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 17

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

### 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 $x,y\in \mathbb {R}$).
    $a^{x+y}=a^{x}\cdot a^{y}\,.$
    $a^{-x}={\frac {1}{a^{x}}}\,.$
    $(a^{x})^{y}=a^{xy}\,.$
    $(ab)^{x}=a^{x}b^{x}\,.$

### Exercise

Prove that for the logarithm to base $b$ the following calculation rules hold.
* 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 $b$.
* We have $\log _{b}(y\cdot z)=\log _{b}y+\log _{b}z$.
* We have $\log _{b}y^{u}=u\cdot \log _{b}y$ for $u\in \mathbb {R}$.
* 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 marks)

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 marks)

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

### Exercise (5 marks)

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 ( marks)

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

### Exercise (4 marks)

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 ( marks)

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 $f(x)=b^{x}$.