Squeeze Theorem

EXAMPLE 18: Determine if the sequence   converges of diverges

EXAMPLE 19: Discuss the convergence of the sequence

Both numerator and denominator approach infinity as   ` but here we have no corresponding function for use with l’Hospital’s Rule (x! is not defined when x is not an integer). Let’s write out a few terms to get a feeling for what happens to  as  gets large:

It appears from these expressions that the terms are decreasing and perhaps approach 0. To confirm this, observe from Equation 8 that

Notice that the expression in parentheses is at most 1 because the numerator is less than (or equal to) the denominator. So

We know that    Therefore   by the Squeeze Theorem

EXAMPLE 20: Use the Squeeze Theorem to determine if the sequence  converges or diverges.

Definition A sequence   is called increasing if  . It is called decreasing if   for all . A sequence is monotonic if it is either increasing or decreasing.

EXAMPLE 21: The sequence  is decreasing because

The right side is smaller because it has a larger denominator.  and so    for all 

EXAMPLE 22: Show that the sequence   is decreasing.

 We must show that   that is,

This inequality is equivalent to the one we get by cross-multiplication:

Since  we know that the inequality   is true. Therefore   is decreasing.

Definition A sequence   is bounded above if there is a number M such that

 It is bounded below if there is a number m such that 

If it is bounded above and below, then   is a bounded sequence.

Monotonic Sequence Theorem Every bounded, monotonic sequence is convergent.

EXAMPLE 23: determine whether or not the sequence    is bounded

the sequence is increasing, it is monotonic and bounded

bounded below  and   bounded above   

EXAMPLE 23: determine whether or not the sequence   is bounded

the sequence is not monotonic and bounded