- Exercises
Let be a real
convergent sequence
and
.
Show that the sequence is also convergent and
-
holds.
Let be a real
convergent sequence
such that
for all
and
.
Show that is also convergent and
-
holds.
Let
and
be real
convergent sequences.
Suppose that
and
for all
.
Show thatis also convergent and that
-
holds.
Let
.
Prove that the sequence converges to .
Let denote the
Heron-sequence
for the computation of with initial value
and let be the Heron-sequence for the computation of with initial value
.
- Compute
and .
- Compute
and .
- Compute
and .
- Does the
product sequence
converge within the rational numbers?
Let
.
For initial value
,
let a real sequence be recursively defined by
-
Show the following statements.
(a) For
,
we have
for all
,
and the sequence is strictly decreasing.
(b) For
,
the sequence is constant.
(c) For
,
we have
for all
.
and the sequence is strictly increasing.
(d) The sequence converges.
(e) The limit is .
Decide whether the
sequence
-
converges
in , and determine, if applicable, its
limit.
Let and be polynomials with
.
Determine, depending on and , whether
-
(which is defined for sufficiently large) is a convergent sequence or not, and determine the limit in the convergent case.
Let
be a non-negative real number and
.
Prove that the sequence defined recursively as
and
-
converges to .
Give an example of a real sequence, that does not converge, but it contains a convergent subsequence.
Suppose that for every natural number , a
null sequence
is given, we denote the -th member of the -th sequence by . Is the sequence , whose -th member is defined by
-
also a null sequence? Is it possible to apply
Lemma 8.1
(1)
in this exercise?
Suppose that for every natural number , a
null sequence
is given, we denote the -th member of the -th sequence by . Is the sequence , whose -th member is defined by
-
also a null sequence? Is it possible to apply
Lemma 8.1
(3)
in this exercise?
Discuss the Cauchy principle of approximation: If, applying a method of approximations, the approximations do not get much better anymore, thought the effort is increased, then the reason is probably that we are close to the truth. Consider mathematical and and non-mathematical examples and counter-examples.
Give an example of a Cauchy sequence in , such that (in ) it does not converge.
We consider the sequence given by
-
Show that this is a null sequence.
Show that the sequence with
converges.
Let be the sequence of the Fibonacci numbers and
-
Prove that this sequence converges in and that its limit satisfies the relation
-
Calculate this .
Hint: Show first with the help of the Simpson formula that it is possible to define a sequence of nested intervals with these fractions.
For two nonnegative
real numbers
and ,
we call
-
their
geometric mean.
Let and be two non-negative real numbers. Prove that the
arithmetic mean
of these numbers is larger than or equal to their
geometric mean.
Let
be a real number. We consider the real sequence
-
(with
).
- Show that the sequence is decreasing.
- Show that all members of the sequence are .
- Show that the sequence
converges
to .
Let , , be a sequence of nested intervals in . Prove that the intersection
-
consists of exactly one point
.
Let , , be a sequence of nested intervals in and let be a real sequence with
for all
.
Prove that this sequence converges to the unique number belonging to the intersection of the family of nested intervals.
Give an example of a sequence of closed intervals
()
-
such that is a null sequence, such that the intersection consists in exactly one point, but such that is not a family of
nested intervals.
Show, using
Bernoullis's inequality,
that the sequence
-
is
increasing.
With a similar argument we can show that the sequence is decreasing and that defines a sequence of nested intervals. The real number defined by these nested intervals is the Euler's number . During this course we will encounter another description for this number.
Let
be a real number. Prove that the sequence , diverges to .
Let be a
real number
with
.
Show that the sequence
converges
to .
Give an example of a real sequence , such that it contains a subsequence that diverges to and also a subsequence that diverges to .
Examine the convergence of the following sequence
-
where
.
Prove that the sequence diverges to .
Let be a real sequence with
for all
.
Prove that the sequence diverges to if and only if the sequence converges to .
- Hand-in-exercises
Determine the
limit
of the real sequence given by
-
Determine the limit of the real sequence given by
-
Give examples of convergent sequences of real numbers
and
with
, ,
and with
such that the sequence
-
- converges to ,
- converges to ,
- diverges.
Decide whether the sequence
-
converges and in case determine the limit.
Examine the convergence of the following real sequence
.
Let
be a
convergent sequence
with limit . Show that the sequence converges to .
Let
be positive real numbers. We define recursively two sequences and such that
,
, and that
-
-
Prove that is a sequence of nested intervals.