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

$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}\text{ and }\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}.$

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 $\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$ and $\sum _{n=0}^{\mathrm{\infty }}{b}_n{x}^n$ be two power series absolutely convergent in $x\in ℝ$. Prove that the Cauchy product of these series is exactly

$\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n\text{ where }{c}_n=\sum _{i=0}^n{a}_i{b}_{n-i}.$

Exercise

Let $x\in ℝ$ , $|x|<1$. Determine (in dependence of $x$) the sum of the two series

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

Exercise

Let

$\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$

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

$\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n={\left(\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n\right)}^3.$

Exercise

Prove that the real function defined by the exponential

$\exp :ℝ⟶ℝ,x⟼\exp x,$

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

$ℝ⟶ℝ,x⟼a^x,$

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

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 $u\in ℝ$.
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 $2\mathrm{\%}$. After what period of time (in years and days), the prices have doubled?

Exercise

Let $b,c>0$. Show that

${\lim }_{b\to 0}{b}^c=0.$

Hand-in-exercises

Exercise (3 marks)

Compute the coefficients $c_0,c_1,\dots ,c_5$ of the power series $\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$, which is the Cauchy product of the geometric series with the exponential series.

Exercise (4 marks)

Let

$\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$

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

$\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n={\left(\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n\right)}^4.$

Exercise (5 marks)

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

$R_{N+1}(x)=\exp x-\sum _{n=0}^N\frac{x^n}{n!}=\sum _{n=N+1}^{\mathrm{\infty }}\frac{x^n}{n!}$

be the remainder of the exponential series. Prove that for

$|x|\le 1+\frac{1}{2}N$

the remainder term estimate

$|R_{N+1}(x)|\le \frac{2}{(N+1)!}|x{|}^{N+1}$

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 $+\mathrm{\infty }$.

Exercise ( marks)

Let

$f:ℝ⟶ℝ$

be a continuous function $\ne 0$, with the property that

$f(x+y)=f(x)\cdot f(y)$

for all $x,y\in ℝ$. Prove that $f$ is an exponential function, i.e. there exists a $b>0$ such that $f(x)=b^x$.