Some Recursively Defined Sequences

A Convergent Sequence

Sequence that a sequence MATH is defined recursively by the equations $a_{1}=3$ and
MATH
It is not hard to prove by mathematical induction that this sequence is increasing and bounded above and that, consequently, it converges. Furthermore, if
MATH
then, letting MATH in the equation MATH we obtain
MATH
Since this equation can be expressed in quadratic form we can solve it easily obtaining MATH

The Fixed Point Method

We know that if MATH for all $x>-5$ then the number $a$ must be a fixed point of this function $f.$ To find this fixed point graphically we draw the graph of $f$ and the line $y=x$ in the same system. Where these two graphs intersect is the required fixed point.
sequence1__15.png

Looking at The Sequence

We look at the equation MATH and calculate a sequence of iterates of the function $f$ starting at the number $3.$ [This is sort of like the 'coweb' method below]. You will obtain


MATH

This table suggests very strongly that the limit of the given sequence is about $2.791288.$

A Divergent Sequence

In this example we look at the sequence MATH defined by $a_{1}=3$ and MATH for each $n.$ 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 $a=\sqrt{5+a}+a^{2}$ has no real solution. If we look for a fixed point of the function $f $ defined by MATH by plotting the graph of $f $ in the same system as the line $y=x$ then we obtain the next figure that shows that the two graphs do not intersect.$x$
sequence1__32.png

Finally, if we define MATH and make a column of iterates then we see compelling evidence that MATH as MATH
MATH

Coweb

A cobweb diagram is a graphical method of looking at the sequence MATH generated by a dynamical system
MATH
This can be demonstrated by the following diagram:


MATH

Newton's Method

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 $(x_{0},f(x_{0}))$ is a point on the graph of the function $f.$ The tangent line at this point is
MATH
The $x-$intercept of this line (by setting $y=0)$ yield,
MATH
Base on this idea, here is the Newton's method:
MATH

Some Exercises

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.

  1. We define $a_{1}=2$ and for each $n\ge 1$ we have
    MATH

  2. We define $a_{1}=-1$ and for each $n\ge 1$ we have
    MATH

  3. We define $a_{1}=1$ and for each $n\ge 1$ we have
    MATH

  4. We define $a_{1}=1$ and for each $n\ge 1$ we have
    MATH

  5. We define $a_{1}=0$ and for each $n\geq 1$ we have
    MATH
    Can you see why the limit of this sequence is MATH

This document created by Scientific WorkPlace 4.0.