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

*Exercises*

Show that .

Let

- Find the smallest such that
- Find the smallest such that

Karl wants to compute approximately with his programmable pocket calculator the value of the series . The display of the calculator shows 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“.

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

Prove that the two series

Let . Prove that the series

diverges.

Prove that the series

Show the estimate

Let

be convergent series of real numbers with sums and . Prove the following statements.

- The series with is convergent with sum equal to .
- For the series with is also convergent with sum equal to .

Prove the Cauchy criterion for series of real numbers.

Let be a real series with for all . Show that this series converges if and only if it is bounded from above.

Give an example of s real series such that it is (as a sequence of partial sums) bounded, but does not converge.

Let . Prove that the series

is convergent.

Prove the following Comparison test:

Let and be two series of non-negative real numbers. The series is divergent and moreover we have for all . Then the series is also divergent.

Decide whether the series

converges.

Let be a
real series.
A
*reordering*
of this series is a series with
for a
bijective mapping

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.

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.

In a shared flat for students, Student 1 prepares coffee and he puts the amount 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 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 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?

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?

We consider the alternating series of the unit fractions with

so

which converges.

a) Show that the reordered series

converges.

b) Give a reordering of the series which diverges.

Let be an absolutely convergent real series. Show that then also every reordering of this series converges to the same limit.

Compute the series

Compute the sum

Two people, and , are in a pub. wants to go home, but still wants to drink a beer. "Well, we just drink another beer, but this is the very last“, says . 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?

Show that

Let . Determine and prove a formula for the series

Let be a real series with for all . Suppose that the sequence of fractions

converge to a real number with . Show, using the ratio test, that the series converges.

Determine whether the following series converge:

- ,
- ,
- .

*Hand-in-exercises*

### Exercise (4 marks)

Determine whether the series

converges.

### Exercise (3 marks)

Compute the sum

### Exercise (3 marks)

Let , . A sequence of digits, given by

(where
)
defines a
real series^{[1]}

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

### Exercise (4 marks)

Into a settling pond with a capacity of , at the beginning of each day, water is filled in. It contains 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 . At the end of each day, 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 clear water?

### Exercise (4 marks)

Prove that the series

### Exercise (5 marks)

The situation in the turtle paradox of Zenon of Elea is the following: a slow turtle (with speed ) has a starting point compared with the faster Achilles (with speed and starting point ). They start at the same time. Achilles can not catch the tortoise: when he arrives at the starting point of the tortoise , the turtle is not there any more, but a little further, let's say at the point . When Achilles arrives at the point , the turtle is once again a bit further at the point , and so on. Calculate the elements of the sequence , the associated time points , 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*

- ↑ So here the index runs in the opposite direction.

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