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

*Exercises*

Compute the first five terms of the Cauchy product of the two convergent series

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.

Let and be two power series absolutely convergent in . Prove that the Cauchy product of these series is exactly

Let , . Determine (in dependence of ) the sum of the two series

We consider the polynomial

- Compute the value of for the points .
- Sketch the graph of on the interval . Does there exist a relation to the exponential function ?
- Determine a zero of within , with an error of at most .

Compute by hand the first digits in the decimal system of

Show the following estimates.

a)

b)

Let denote a positive real number. Prove that the exponential function

fulfills the following properties.

- We have for all .
- We have .
- For and , we have .
- For and , we have .
- For , the function is strictly increasing.
- For , the function is strictly decreasing.
- We have for all .
- For , we have .

Let

be a continuous function , with the property that

for all . Prove that is an exponential function, i.e. there exists a such that .

Show that an exponential function

transforms an arithmetic mean into a geometric mean.

Let

be an exponential function with . For every , the line defined by the two points and has an intersection point with the -axis, which we denote by . Show

Sketch the situation.

Give an example of a continuous, strictly increasing function

fulfilling and for all , which is different from .

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

Let . Show that the power function

is continuous.

Let be a positive real number and . Show that the number defined by

is independent of the fraction representation of .

Let and let be a rational number. Show that the expression

is compatible with the definition

Compute

up to an error of .

Compute

up to an error of .

Compare the two numbers

Compare the thee numbers

Let . Show that

Let . Show that

holds.

Decide whether the real sequence

(with ) converges in , and determine, if applicable, the limit.

Prove that for the logarithm to base the following calculation rules hold.

- We have and , ie, the logarithm to base is the inverse to the exponential function to the base .
- We have .
- We have for .
- We have

*Hand-in-exercises*

### Exercise (4 marks)

Compute , using the
exponential series,
so that the error is at most .

The estimate on the remainder from
Exercise 12.29
can be used.

### Exercise (3 marks)

Compute the coefficients of the power series , which is the Cauchy product of the geometric series with the exponential series.

### Exercise (4 marks)

### Exercise (5 marks)

For and let

be the remainder of the exponential series. Prove that for

the remainder term estimate

holds.

### Exercise (4 marks)

Prove that the real exponential function defined by the exponential series has the property that for each the sequence

diverges to .^{[1]}

### 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 . Unfortunately, the professor looses a percentage of of his cleverness each year.

- Show that after three years of studies, the professor is still more clever that his students.
- 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 . After what period of time (in years and days), the prices have doubled?

*Footnotes*

- ↑ Therefore we say that the exponential function grows faster than any polynomial function.

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