Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 9

Exercises

Exercise

Show that ${}\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}={\frac {1}{2}}$.

Exercise

Let

${}x_{n}:=\sum _{k=1}^{n}{\frac {1}{k}}\,.$

1. Find the smallest ${}n$ such that

   ${}x_{n}\geq 2\,.$

2. Find the smallest ${}n$ such that

   ${}x_{n}\geq 2{,}5\,.$

Exercise

Karl wants to compute approximately with his programmable pocket calculator the value of the series ${}\sum _{n=1}^{\infty }{\frac {1}{n}}$. The display of the calculator shows ${}8$ positions after the decimal point. The calculator performs one addition (meaning a member of the series is added to the previous sum) per second and shows the result. Karl has programmed correctly and considers the following strategy: "If the display does not change for a complete hour, then we are probably close to the true result and we accept this as a good approximation. Then the calculator shall stop“.

1. What is roughly the last member to be added?
2. What do you think about this strategy?

Exercise

Prove that the two series

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

are divergent.

Exercise

Let ${}a,b\in \mathbb {R} _{+}$. Prove that the series

$\sum _{k=0}^{\infty }{\frac {1}{ak+b}}$

diverges.

Exercise

Prove that the series

$\sum _{k=1}^{\infty }{\frac {1}{\sqrt {k}}}$

diverges.

Exercise

Show the estimate

${}\sum _{i=1}^{n}{\frac {1}{\sqrt {i}}}\leq 3{\sqrt {n}}\,.$

Exercise

Let

$\sum _{k=0}^{\infty }a_{k}{\text{ and }}\sum _{k=0}^{\infty }b_{k}$

be convergent series of real numbers with sums ${}s$ and ${}t$. Prove the following statements.

1. The series ${}\sum _{k=0}^{\infty }c_{k}$ with ${}c_{k}=a_{k}+b_{k}$ is convergent with sum equal to ${}s+t$.
2. For ${}r\in \mathbb {R}$ the series ${}\sum _{k=0}^{\infty }d_{k}$ with ${}d_{k}=ra_{k}$ is also convergent with sum equal to ${}rs$.

Exercise *

Prove the Cauchy criterion for series of real numbers.

Exercise

Let ${}\sum _{k=0}^{\infty }a_{k}$ be a real series with ${}a_{k}\geq 0$ for all ${}k$. Show that this series converges if and only if it is bounded from above.

Exercise

Give an example of s real series ${}\sum _{k=0}^{\infty }a_{k}$ such that it is (as a sequence of partial sums) bounded, but does not converge.

Exercise

Let ${}k\geq 2$. Prove that the series

$\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}$

is convergent.

Exercise

Prove the following Comparison test:

Let ${}\sum _{k=0}^{\infty }a_{k}$ and ${}\sum _{k=0}^{\infty }b_{k}$ be two series of non-negative real numbers. The series ${}\sum _{k=0}^{\infty }b_{k}$ is divergent and moreover we have ${}a_{k}\geq b_{k}$ for all ${}k\in \mathbb {N}$. Then the series ${}\sum _{k=0}^{\infty }a_{k}$ is also divergent.

Exercise

Decide whether the series

$\sum _{n=1}^{\infty }{\frac {n!}{n^{n}}}$

converges.

Let ${}\sum _{k=0}^{\infty }a_{k}$ be a real series. A reordering of this series is a series ${}\sum _{k=0}^{\infty }b_{k}$ with ${}b_{k}=a_{\sigma (k)}$ for a bijective mapping

${}\sigma \colon \mathbb {N} \rightarrow \mathbb {N}$.

In the case of a reordering of a series, the same summands occur. However, the corresponding sequence of the partial sums has changed and maybe also convergence.

Exercise

Show that in a real sequence if you change finitely many sequence elements then neither the convergence nor the limit changes, and that in a series if you change a finite number of series terms then the convergence does not change, but the sum changes.

Exercise

In a shared flat for students, Student 1 prepares coffee and he puts the amount ${}x_{1}$ of coffee in the coffee filter. Then Student 2 looks horrified and says: "Do you want us all to be already completely awake“? and he takes the amount of coffee ${}x_{2}<x_{1}$ back out of the filter. Then Student 3 comes and says: "Am I in a flat of sissies“? and he puts back an amount of coffee ${}x_{3}<x_{2}$ in it. So it goes on indefinitely, alternating between putting in and taking out smaller amounts of coffee from the coffee filter. How can one characterize if the amount of coffee in the filter converges?

Exercise

Since on the previous day the coffee has become too weak for the supporters of a strong coffee, they develop a new strategy: they want to get up early, so that at the beginning of the day and between every two supporters of a weak coffee (who take out coffee from the coffee machine) there are always two supporters of a strong coffee putting in coffee. The amount of coffee that each person puts in or takes out does not change, and also the order in both camps does not change. Can they make the coffee stronger with this strategy?

Exercise

We consider the alternating series of the unit fractions ${}\sum _{n=1}^{\infty }x_{n}$ with

${}x_{n}=(-1)^{n+1}{\frac {1}{n}}\,,$

so

$1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+{\frac {1}{7}}-{\frac {1}{8}}+{\frac {1}{9}}\cdots ,$

which converges.

a) Show that the reordered series

$1+{\frac {1}{3}}-{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{11}}-{\frac {1}{6}}\cdots ,$

converges.

b) Give a reordering of the series which diverges.

Exercise

Let ${}\sum _{k=0}^{\infty }a_{k}$ be an absolutely convergent real series. Show that then also every reordering of this series converges to the same limit.

Exercise

Compute the series

${}1+{\frac {1}{5}}+{\frac {1}{25}}+{\frac {1}{125}}+{\frac {1}{625}}+\ldots \,.$

Exercise

Compute the sum

$\sum _{n=3}^{\infty }{\left({\frac {2}{5}}\right)}^{n}.$

Exercise

Two people, ${}A$ and ${}B$, are in a pub. ${}A$ wants to go home, but ${}B$ still wants to drink a beer. "Well, we just drink another beer, but this is the very last“, says ${}A$. Then B wants another beer, but since the previous beer was definitely the last one, they agree to drink a last half beer. After that they drink a last quarter of a beer, and then the last eighth of beer, and so on. How many "very last beer“ do they drink overall?

Exercise

Show that

${}\sum _{n=0}^{\infty }{\frac {3^{n}+(-2)^{n}}{5^{n}}}={\frac {45}{14}}\,.$

Exercise

Let ${}z\in \mathbb {R} ,\,\vert {z}\vert <1$. Determine and prove a formula for the series

$\sum _{k=0}^{\infty }(-1)^{k}z^{k}.$

Exercise

Let ${}\sum _{n=0}^{n}a_{n}$ be a real series with ${}a_{n}\neq 0$ for all ${}n$. Suppose that the sequence of fractions

${}x_{n}={\frac {a_{n+1}}{a_{n}}}\,$

converge to a real number ${}x$ with ${}-1<x<1$. Show, using the ratio test, that the series converges.

Exercise

Determine whether the following series converge:

1. ${}\sum _{n=1}^{\infty }{\frac {n+3}{n^{3}-n^{2}-n+2}}$,
2. ${}\sum _{n=1}^{\infty }{\frac {n+{\sqrt {n}}}{n^{2}-{\sqrt {n}}+1}}$,
3. ${}\sum _{n=1}^{\infty }(-1)^{n}({\sqrt {n}}-{\sqrt {n-1}})$.

Hand-in-exercises

Exercise (4 marks)

Determine whether the series

$\sum _{n=0}^{\infty }{\frac {2n+7}{n^{3}-4n^{2}+3n-5}}$

converges.

Exercise (3 marks)

Compute the sum

$\sum _{n=3}^{\infty }{\left({\frac {2}{3}}\right)}^{n}.$

Exercise (3 marks)

Let ${}g\in \mathbb {N}$ , ${}g\geq 2$. A sequence of digits, given by

$z_{i}\in \{0,1,\ldots ,g-1\}{\text{ for }}i\in \mathbb {Z} ,\,i\leq k,$

(where ${}k\in \mathbb {N}$) defines a real series

$\sum _{i=k}^{-\infty }z_{i}g^{i}.$

Prove that such a series converges to a unique non-negative real number.

Exercise (4 marks)

Into a settling pond with a capacity of ${}2000\,\mathrm {m} ^{3}$, at the beginning of each day, ${}200\,\mathrm {m} ^{3}$ water is filled in. It contains ${}10\,\%$ of a certain contaminant, and is mixed completely with the water. During each day, by a biological process, the given amount of contaminant is reduced by ${}20\,\%$. At the end of each day, ${}200\,\mathrm {m} ^{3}$ water leaves the pond. Which percentage of the contaminant will there be in the drain, in the long run, if the pond is filled at the beginning with ${}1800\,\mathrm {m} ^{3}$ clear water?

Exercise (4 marks)

Prove that the series

$\sum _{n=0}^{\infty }{\frac {n^{2}}{2^{n}}}$

converges.

Exercise (5 marks)

The situation in the turtle paradox of Zenon of Elea is the following: a slow turtle (with speed ${}v>0$) has a starting point ${}s>0$ compared with the faster Achilles (with speed ${}w>v$ and starting point ${}0$). They start at the same time. Achilles can not catch the tortoise: when he arrives at the starting point of the tortoise ${}s_{0}=s$, the turtle is not there any more, but a little further, let's say at the point ${}s_{1}=s_{0}$. When Achilles arrives at the point ${}s_{1}$, the turtle is once again a bit further at the point ${}s_{2}>s_{1}$, and so on. Calculate the elements of the sequence ${}s_{n}$, the associated time points ${}t_{n}$, and the respective limits. Compare these limits with the distance point where Achilles catches the turtle (you can calculate it directly from the given data).

Footnotes

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