6 Sequences

Abbott provides us with the following definition for a sequence [1].

Definition I.12
A sequence is a function whose domain is .

So, a sequence defined by a function is really just a number pattern where is the th term in the pattern. In calculus, we are interested in whether the sequence converges or diverges (does not converge).

One way to find whether a sequence defined by converges is to find

If this limit exists, then the sequence not only converges but converges to that limit [15].

We also have a theorem that can be used to determine if a sequence converges when it is hard to find the limit or to know if it even exists. The theorem is stated by Larson and Edwards as follows [15].

Theorem I.13

If a sequence is bounded and monotonic, then it converges.

This theorem raises an important question. What do bounded and monotonic mean?

A sequence is bounded above when is less than or equal to a real number for all and bounded below when is greater than or equal to for all . A sequence is bounded when it is bounded above and below [15].

Monotonic means that either

for all [15]. That is, the sequence is either nondecreasing or nonincreasing.

We will not go through a formal proof here, but we will discuss why this theorem makes sense. Consider a nondecreasing, bounded sequence . That is,

where is a real number. Every sequence must either converge or diverge, and a nondecreasing sequence diverges if and only if for every real number we can find such that . Because there exists a real number such that for all , does not diverge and must converge. Thus, if a sequence is nondecreasing and bounded, it must converge. Similar reasoning can be used for a nonincreasing, bounded sequence.

Let’s now look at a problem that demonstrates how to prove a sequence is monotone and bounded so that the theorem can be applied. This is a problem I completed for Calculus II and comes from Larson and Edwards [15].

Example 14
Does converge?

Solution
The terms of are

The terms appear to be decreasing but will the sequence continue this way? Because is positive, we have that for all

Thus, is decreasing and consequently monotonic. Moreover, is bounded above by 13.

Now, we need to determine if is bounded below. Because is a natural number, we have

Thus, is bounded below by 8. By Theorem I.13, converges.

We are now going to begin our discussion of series, which builds upon the ideas of sequences.

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