Welcome back to Logic Language and Information 2 and this is
the fourth in the series of lessons on mathematics and logic.
Beginning with the basic definition any infinite
list of real numbers is called a sequence.
We denote sequences by expressions such a sub n
in parentheses indexed by numbers n in the natural numbers.
The index n, identifies the position of the term a sub n in the sequence.
We've already seen finite sequences in the
course of developing the semantics of predicate logic.
For a k-place predicate, we use length-k sequences of objects,
or k-tuples of objects to describe the meaning of the predicate.
In contrast an infinite sequence is a list
that keeps on going and going and going forever.
Where the entries an are generated according to some rule.
Most famous sequence is the Fibonacci sequence.
Start with a sub zero is 1, and a sub one is 1.
And then continue according to the recursive rule that a sub n plus 2
is equal to the sum of a sub n plus 1 and a sub n.
So, we start a sub 2 is 2, one plus one is two, a sub 3 is
3, a sub 4 is 5, a sub 5 is 8, a sub 6 is 13, and so on.
The natural world exhibits Fibonacci numbers in multiple ways, such as the
number in multiple ways, such as the number of petals on a
flower, the number of, the growth patterns of branches from the main
stem of a plant, the patterns of scales in a pine cone.
As a mathematical sequence, its entries keep on getting
larger and larger without bound, it goes to infinity.
In contrast, some sequences of real numbers get closer and
closer to a fixed number, they converge to a limit.
One of the first examples we all usually see of
an infinite repeat, is of an infinite repeating decimal number.
0.999 repeated.
Now, the sequence is generated by the rule a sub n is the fraction.
10 to the power N minus one over ten to the power N.
So we end up with 0, 0.9, 0.99, 0.999, 0.9999, and so on.
For
this
sequence it is clear that it is heading towards a limit of.
0.9 infinitely repeated.
Which we can indicate using an over line on the 9, like this.
Now, if we applied a property, called label Roman 3
of the distance is less than predicate from lesson five three.
Which expresses that the distance, if the distance is less
than every positive number exactly when two numbers are identical.
Then so we can conclude that zero point nine repeated is in fact identical.
To the number 1, because there is no positive distance epsilon between them.
So back to the previous lesson 5, 1,
which is just to Archimedean, approximation of pi.
This is an approximation sequence for inscribed or n side polygons.
Illustrated here goes as follows.
So, we're actually working with a circle of radius one and diameter two.
So that pi comes out as equal to half the perimeter, or circumference.
At stage zero, we start with a four sided polygon, a
square, each of whose sides are length the square root of two.
This comes from using Pythagoras' theorem for right angle triangles.
So P sub zero is one half of four times the square root of two.
Which comes out as 2 root 2.
Then proceeding at each successive stage by
doubling the number of sides so progressing
to 6, 16, 32 sides and so on and using Geometry particularly Pythagoras' theorem.
To calculate the length of the sides of the stage in polygons recursively.
Now, we can evaluate, the side length, l sub n plus 1.
As equal to the square root of 2 minus the square
root of 4 minus l sub n, all squared.
[LAUGH] This approximation, is accurate to two
decimal places by state and equals three.
When the polygon has thirty two sides, this is
one more stage than is illustrated in the image.
But try to draw it and it gets too
close to the circle and you can't just differentiate, okay?
Another quite differently behaved sequence, comes from
different formulas for a sub n, depending
upon whether n is an even natural number, or n is a odd natural number.
So, for example, suppose a sub n is 1 over n plus 1 if n
is even, and, so that's n is 0, 2, 4, 6, and so
on, and suppose that a sub n is minus 1 over n
plus 1 if n is odd, so n is 1, 3, 5, 7, and so on.
So we get the sequence shown in the
table, where the entries alternate between positive and negative.
But are still getting closer and closer in distance
to the value zero, whilst never quite getting there.
Okay, one more example, to lead you into an exercise.
Consider the sequence starting with a sub zero is zero, and a
sub one is one, and then continuing according to the recursive rule.
It says a sub n plus two is the average of the previous two entries.
a sub n plus 1 and a sub n.
So, the average is just the midpoint calculated by taking one half of the sum.
The resulting sequence is a repeated average, or moving average
sequence.
This sequence begins with 0, then 1, 0.5, 0.75.
0.625, 0.6875, 0.65625, 0.671875, [SOUND] and so on, As written with gone
to the first ten
entries and
we get up to eight decimal places in those ten places.
Now the, this repeated averages recursive rule plus the values for a sub 0 and
a sub 1 together constitute what's called
a second order difference equation with initial conditions.
From the general theory of difference equations and your taking this
on faith we're not doing mathematics as such here, but rather logic.
This class of equation can be solved to give an explicit solution.
The general nth term a-sub-n is equal to
2 3rds of 1 minus 1 over 2 to the power n where n is even,
and 2 3rds of 1 plus 1 over 2 to the power n where n is odd.
Now, just from this explicit solution formula,
which of the following options seems correct?
Now, at this point, don't stress, we will formally
define convergence and limits of sequences in the next lesson.
This exercise is just to get you thinking about it,
okay?
So how did you go?
The correct answer is B.
The entries are getting closer and closer in distance to the number two thirds.
Even though the entries alternate between being a little bit above and.
This target number of 2 3rds.
Recall the distances less than predicate expressing that numbers x
and y are less than epsilon or part in distance.
We're going to be using this 3-place
predicate D together with quantifiers to precisely define
what we mean by an infinite sequence of real numbers converging to a limit, R.
The epsilon in the third place of the atomic formula.
D A sub n epsilon will give a bound on the accuracy or error margin.
Expressing how close the term A sub n is to this target number, r.
We also need to consider smaller and smaller values of epsilon, to capture
the idea that we are getting closer and closer to the limit number.
For example, we could take epsilon in progression as one out of ten, one out of
a hundred, one out of a 1000, one over 10000 and so on and
the patent one over 10 to the power of K as natural number K goes to infinite.
This brings us the end of this lesson on sequences of real numbers and in the next
and final lesson in this series we will arrive at our goal and rigorously define.
Convergence and divergence of sequences of real numbers in pretty good logic.
So see you then.