Programmed tutorial: Sequence and the Squeeze Theorem

It is called “n factorial”.

How is it defined?

Use this definition to determine and plot the terms of this sequence on paper. Then check your diagram by clicking “Next”.

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
(Even though some of the points appear to be on the x-axis, they are really slightly above the axis.)

To determine convergence, can we use the method of examining the limit of a related function of x, where x is a real number?

No.

Why?

n! is not defined for the set of real numbers.

What method can we use?

We can use the Squeeze Theorem for Sequences.

State this theorem.

Let {a_n}, {b_n} and {c_n} be sequences such that

a_n < c_n < b_n for all n greater than some positive integer N, and

then

Using the notation ,
what does the graph suggest we use for a_n?

Since the terms seem to approach 0, we can use a_n = 0.

For b_n, we need a term which is larger than
Should we use a bigger or smaller denominator for b_n?

A smaller denominator would give us a larger term
for b_n.

Let’s try

Add the points for b_n to your earlier diagram.  Check by clicking “Next”.

alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag! We see that the first two red points replace the blue ones, and the remaining red points are above the blue.  This is consistent with the theorem: that b_n be greater than, or equal to c_n.

Collect these ideas.

We’ll squeeze between 0 and

Write this as an inequality.

becomes

What is the limit of a_n?

What is the limit of  b_n?

What is our conclusion?

We conclude
That is, the sequence converges, with a limit of 0.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor