- 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
-
Let
-
be an absolutely convergent power series. Compute the coefficients of the powers
in the third power
-
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
Compute , using the
exponential series,
so that the error is at most .
The estimate on the remainder from
Exercise 12.29
can be used.
Compute the coefficients of the power series , which is the
Cauchy product
of the
geometric series
with the
exponential series.
Let
-
be an absolutely convergent power series. Determine the coefficients of the powers
in the fourth power
-
For
and
let
-
be the remainder of the exponential series. Prove that for
-
the remainder term estimate
-
holds.
Prove that the real exponential function defined by the exponential series has the property that for each
the sequence
-
diverges to .[1]
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.
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.