18. Sequences

b1. Limits - Definition

The limit of the sequence \(a_n\), denoted \(\lim\limits_{n\to\infty}a_n\), tells what number the terms are approaching as we go further and further out in the sequence (as \(n\) gets arbitrarily large).

When we are given a sequence, the typical questions we ask are:

1. Does the sequence have a limit?
2. If so, what is the limit of the sequence?
3. If not, is the limit infinite, (i.e. \(+\infty\) or \(-\infty\)) or oscillatory divergent (i.e. neither \(+\infty\) nor \(-\infty\))?

There is a precise definition of the limit of a squence, but it is beyond the scope of most classes. If you want to read about it, it is at the end of this chapter. However, it is sufficient to understand the following intuitive definition:

The sequence \(a_n\) has a limit if the terms \(a_n\) get closer and closer to a finite number \(L\) as \(n\) gets arbitrarily large. In that case, we say \(\lim\limits_{n\to\infty}a_n\) exists and is equal to \(L\), and we write: \[\lim_{n\to\infty}a_n=L \] If a sequence has a limit \(L\), we say it is convergent and that it converges to \(L\).
If a sequence does not have a limit, we say it is divergent or that it diverges.

If the sequence diverges, it may still have an infinite limit.

If the terms \(a_n\) get arbitrarily large and positive as \(n\) gets arbitrarily large, then we say the sequence diverges to positive infinity (\(+\infty\)) or that the limit is \(+\infty\) and write: \[\lim_{n\to\infty}a_n=\infty \] If the terms \(a_n\) get arbitrarily large and negative as \(n\) gets arbitrarily large, then we say the sequence diverges to negative infinity (\(-\infty\)) or that the limit is \(-\infty\) and write: \[\lim_{n\to\infty}a_n=-\infty \] If the sequence is divergent but does not diverge to either \(\infty\) or \(-\infty\), then we say it is oscillatory divergent.

To say that the limit is plus or minus infinity does not say that the limit exists! It merely says the way in which it does not exist, i.e. the way in which it diverges.

It is certainly possible to diverge without diverging to \(\infty\) or \(-\infty\).

Plots are insufficient to determine a limit. However, examples and exercises appear on the next page.