Sequence that a sequence
is defined recursively by the equations
and
It is not hard to prove by mathematical induction that this sequence is
increasing and bounded above and that, consequently, it converges.
Furthermore, if
then, letting
in the equation
we obtain
Since this equation can be expressed in quadratic form we can solve it easily
obtaining
We know that if
for all
then the number
must be a fixed point of this function
To find this fixed point graphically we draw the graph of
and the line
in the same system. Where these two graphs intersect is the required fixed
point.
We look at the equation and calculate a sequence of iterates of the function starting at the number [This is sort of like the 'coweb' method below]. You will obtain
This table suggests very strongly that the limit of the given sequence is about
In this example we look at the sequence
defined by
and
for each
It is easy to see that this sequence is increasing and that it is unbounded
above. Consistent with this observation is the fact that the equation
has no real solution. If we look for a fixed point of the function
defined by
by plotting the graph of
in the same system as the line
then we obtain the next figure that shows that the two graphs do not
intersect.
Finally, if we define
and make a column of iterates then we see compelling evidence that
as
A cobweb diagram is a graphical method of looking at the sequence
generated by a dynamical system
This
can be demonstrated by the following diagram:
Newton's method is based on the observation that the tangent line is a good
local approximation to to the graph of a function. If
is a point on the graph of the function
The tangent line at this point is
The
intercept
of this line (by setting
yield,
Base
on this idea, here is the Newton's method:
For each of the following recursively defined sequences, show that the sequence is monotone, try to decide whether or not it is bounded, try to find the limit yourself and then find the limit.
We define
and for each
we have
We define
and for each
we have
We define
and for each
we have
We define
and for each
we have
We define
and for each
we have
Can
you see why the limit of this sequence is