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

Exercises

Exercise Create referencenumber

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 Create referencenumber

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 Exercise 12.3

change===

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 Create referencenumber

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 Create referencenumber

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 Create referencenumber

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 Create referencenumber

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

$\exp 1.$

Exercise Create referencenumber

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 Exercise 12.9

change===

Let ${}b$ denote a positiveMDLD/positive (R) real number.MDLD/real number Prove that the exponential functionMDLD/exponential function (base)

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

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.MDLD/strictly increasing (R)
6. For ${}b<1$, the function ${}f$ is strictly decreasing.MDLD/strictly decreasing (R)
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 Create referencenumber

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 Create referencenumber

Show that an exponential functionMDLD/exponential function (R base)

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

transforms an arithmetic meanMDLD/arithmetic mean into a geometric mean.MDLD/geometric mean

Exercise Create referencenumber

Let

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

be an exponential functionMDLD/exponential function (R base) 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 Create referencenumber

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 Create referencenumber

Show that the compositionMDLD/composition of two exponential functionsMDLD/exponential functions (R) is not necessarily an exponential function.

Exercise Create referencenumber

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

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

is continuous.MDLD/continuous (R)

===Exercise Exercise 12.16

change===

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 Exercise 12.17

change===

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

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

is compatible with the definition

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

Exercise Create referencenumber

Compute

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

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

Exercise Create referencenumber

Compute

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

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

Exercise Create referencenumber

Compare the two numbers

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

Exercise Create referencenumber

Compare the thee numbers

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

Exercise Create referencenumber

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

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

Exercise Create referencenumber

Let ${}b>0$. Show that

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

holds.

Exercise Create referencenumber

Decide whether the real sequenceMDLD/real sequence

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

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

===Exercise Exercise 12.25

change===

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) Create referencenumber

Compute ${}e^{3}$, using the exponential series,MDLD/exponential series (R) 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) Create referencenumber

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 productMDLD/Cauchy product (R) of the geometric seriesMDLD/geometric series (R) with the exponential series.MDLD/exponential series (R)

Exercise (4 marks) Create referencenumber

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) Exercise 12.29

change===

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) Create referencenumber

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) Create referencenumber

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) Create referencenumber

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

Footnotes

<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF)