Some Recursively Defined Sequences

A Convergent Sequence

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

The Fixed Point Method

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.

Looking at The Sequence

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

A Divergent Sequence

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

Coweb

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

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:

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 and for each we have

2. We define and for each we have

3. We define and for each we have

4. We define and for each we have

5. We define and for each we have

Can you see why the limit of this sequence is

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