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 ${}x,y\in \mathbb {R}$).

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