- Exercises
Have you ever seen in a newspaper an expression of the form
(a vinculum to indicate a repeating decimal)
or
? Why not? What kind of decimal expansions occur in a newspaper
(in a manual, as a result of a measurement, in a technical description)?
We want to find a rational approximation for the number
, such that the deviation of the true value should be at most
. How good has to be an approximation for
such that we obtain the asked for approximation?
Is there any relation between the Din-Norm for paper and square roots?
Compute by hand the approximations
in Heron's method to find the square root of
with initial value
.
Do the first three steps in Heron's method to compute the square root of
with initial value
(so the approximations
for
shall be computed; these numbers have to be given as fractions reduced to lowest terms).
Let
be a positive real number and let
be the
Heron-sequence
for the computation of
with the initial value
.
Let
,
,
and let
be the Heron-sequence for the computation of
with initial value
. Show that
-

holds for all
.
Determine for the sequence
-

and
-

for which
(minimal)
the estimate
-

holds.
Let
be a real sequence. Prove that the sequence converges to
if and only if for all
a natural number
exists, such that for all
the estimation
holds.
Negate the statement that a sequence
converges
in
to
by transforming the quantifiers.
Examine the convergence of the following sequence
-

where
.
Regarding the sequence
,
somebody says: "The numerator and the denominator are both going to infinity. However, the denominator is much faster, therefore the sequence converges to
“. What do you think about this argument?
Determine whether the following subsets
are
bounded
or not.
-
,
-
,
-
,
-
,
-
,
-
,
-
,
-
,
-
.
Let
be a
real number.
Show that the sequence
is not
bounded.
Let
be a null sequence and let
be a bounded real sequence. Show that also the product sequence
is a null sequence.
Let
and
be convergent real sequences with
for all
.
Prove that
holds.
Let
and
be three real sequences. Let
for all
and
and
be convergent to the same limit
. Prove that also
converges to the same limit
.
Let

be a convergent sequence of real numbers with limit equal to

. Prove that also the sequence
-
converges, and specifically to

.
Let the sequence
be given by
-

- Determine
and
.
- Does this sequence converge in
?
Prove by induction the Binet formula for the Fibonacci numbers. This says that
-

holds
(
).
- Hand-in-exercises
===Exercise (3 marks) ===
Compute by hand the approximations
in Heron's method to find the square root of
with initial value
.
Write a computer-program
(pseudocode)
for the computation of rational approximations for the square root of a rational number using
Heron's method.
- The computer has as many memory units as needed, which can contain natural numbers.
- It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.
- It can add the content of two memory units and write the result into another memory unit.
- It can multiply the content of two memory units and write the result into another memory unit.
- It can print contents of memory units and it can print given texts.
The initial configuration is
-
with
.
Here,
is the number from which we want to compute the square root,
is the initial value and
is the wished-for accuracy. The program shall compute and print the Heron-sequence
(the numerators and denominators are printed successively)
and it shall stop when the member
printed last fulfills the property
-

Attention! All operations are to be done within

!
Determine for the sequence
-

and for
-

for which
(minimal)
the estimate
-

holds.
Let
be a
convergent
real sequence
with
limit
. Show that the sequence defined by
-

also converges to
.
Hint: reduce to the case
.
Prove that the real sequence
-
converges to

.
Hint: Find a suitable estimate for

using the binomial theorem.
Let
and
be sequences of real numbers and let the sequence
be defined as
and
.
Prove that
converges if and only if
and
converge to the same limit.