Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 12

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

We consider the polynomial

${}P=1+X+{\frac {1}{2}}X^{2}+{\frac {1}{6}}X^{3}\,.$

1. Compute the value of ${}P$ for the points ${}-2,-1,0,1,2$.
2. Sketch the graph of ${}P$ on the interval ${}[-2,2]$. Does there exist a relation to the exponential function ${}e^{x}$?
3. Determine a zero of ${}P$ within ${}[-2,2]$, with an error of at most ${}{\frac {1}{4}}$.

Exercise

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

$\exp 1.$

Exercise

Show the following estimates.

a)

${}{\binom {n}{k}}\cdot {\frac {1}{n^{k}}}\leq {\frac {1}{k!}}\,,$

b)

${}{\left(1+{\frac {1}{n}}\right)}^{n}\leq \sum _{k=0}^{n}{\frac {1}{k!}}\,.$

Exercise

Let ${}b$ denote a positive real number. Prove that the exponential function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto b^{x},$

fulfills the following properties.

1. We have ${}b^{x+x'}=b^{x}\cdot b^{x'}$ for all ${}x,x'\in \mathbb {R}$.
2. We have ${}b^{-x}={\frac {1}{b^{x}}}$.
3. For ${}b>1$ and ${}x>0$, we have ${}b^{x}>1$.
4. For ${}b<1$ and ${}x>0$, we have ${}b^{x}<1$.
5. For ${}b>1$, the function ${}f$ is strictly increasing.
6. For ${}b<1$, the function ${}f$ is strictly decreasing.
7. We have ${}(b^{x})^{x'}=b^{x\cdot x'}$ for all ${}x,x'\in \mathbb {R}$.
8. For ${}a\in \mathbb {R} _{+}$, we have ${}(ab)^{x}=a^{x}\cdot b^{x}$.

Exercise

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

Exercise

Show that an exponential function

$\mathbb {R} \longrightarrow \mathbb {R} _{+},x\longmapsto b^{x},$

transforms an arithmetic mean into a geometric mean.

Exercise

Let

${}f(x)=a^{x}\,$

be an exponential function with ${}a\neq 1$. For every ${}x\in \mathbb {R}$, the line defined by the two points ${}(x,f(x))$ and ${}(x+1,f(x+1))$ has an intersection point with the ${}x$-axis, which we denote by ${}s(x)$. Show

${}s(x+1)=s(x)+1\,.$

Sketch the situation.

Exercise

Give an example of a continuous, strictly increasing function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} _{+}$

fulfilling ${}f(0)=1$ and ${}f(x+1)=2f(x)$ for all ${}x\in \mathbb {R}$, which is different from ${}2^{x}$.

Exercise

Show that the composition of two exponential functions is not necessarily an exponential function.

Exercise

Let ${}u\in \mathbb {R}$. Show that the power function

$f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x^{u},$

is continuous.

Exercise

Let ${}b$ be a positive real number and ${}q=n/m\in \mathbb {Q}$. Show that the number defined by

${}b^{q}:={\left(b^{n}\right)}^{1/m}\,$

is independent of the fraction representation of ${}q$.

Exercise

Let ${}a>0$ and let ${}q={\frac {r}{s}}$ be a rational number. Show that the expression

${}a^{q}={\sqrt[{s}]{a^{r}}}\,$

is compatible with the definition

${}a^{q}=\exp(q\ln a)\,.$

Exercise

Compute

$2^{\frac {9}{10}}$

up to an error of ${}{\frac {1}{10}}$.

Exercise

Compute

$5^{\frac {3}{7}}$

up to an error of ${}{\frac {1}{10}}$.

Exercise

Compare the two numbers

${\sqrt {3}}^{-{\frac {9}{4}}}\,\,{\text{ and }}\,\,{\sqrt {3}}^{-{\sqrt {5}}}.$

Exercise

Compare the thee numbers

$2^{\sqrt {3}},\,4,\,3^{\sqrt {2}}.$

Exercise

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

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

Exercise

Let ${}b>0$. Show that

${}\operatorname {lim} _{d\rightarrow 0}\,b^{d}=1\,$

holds.

Exercise

Decide whether the real sequence

${}x_{n}={\frac {5n^{\frac {3}{2}}+4n^{\frac {4}{3}}+n}{7n^{\frac {5}{3}}+6n^{\frac {3}{2}}}}\,$

(with ${}n\geq 1$) converges in ${}\mathbb {R}$, and determine, if applicable, the limit.

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

   ${}\log _{a}y=\log _{a}(b^{\log _{b}y})=\log _{b}y\cdot \log _{a}b\,.$

Hand-in-exercises

Exercise (4 marks)

Compute ${}e^{3}$, using the exponential series, so that the error is at most ${}{\frac {1}{1000}}$.
The estimate on the remainder from Exercise 12.29 can be used.

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 (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 (2 (1+1) marks)

At the begin of the university studies, Professor Knopfloch is double as clever as the students. Within one year of studies, the students are getting more clever by a percentage of ${}10\%$. Unfortunately, the professor looses a percentage of ${}10\%$ of his cleverness each year.

1. Show that after three years of studies, the professor is still more clever that his students.
2. Show that after four years of studies, the students are more clever than the professor.

Exercise (2 marks)

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

Footnotes

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