(problem 1) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

We determine if a sequence converges or diverges.

Some examples of sequences are

The question we will have about a given sequence is if it has a limit.

A sequence has a limit , written if the terms of the sequence get arbitrarily close to
as increases without bound.

If a sequence has a (finite) limit, we say that the sequence converges. Otherwise, we say that it diverges.

If a sequence has a (finite) limit, we say that the sequence converges. Otherwise, we say that it diverges.

Does the sequence converge or diverge?

We compute Since as , we can conclude that and so the sequence converges to . The diagram below shows the terms of the sequence approaching .

We compute Since as , we can conclude that and so the sequence converges to . The diagram below shows the terms of the sequence approaching .

The sequence in the previous problem could be written as .

In general, a sequence of the form converges if and diverges if .

In general, a sequence of the form converges if and diverges if .

Does the sequence converge or diverge?

We compute Since oscillates between and we can conclude that and so the sequence diverges. The diagram below shows the oscillation of the terms of the sequence.

We compute Since oscillates between and we can conclude that and so the sequence diverges. The diagram below shows the oscillation of the terms of the sequence.

Does the sequence converge or diverge?

Since we conclude that the sequence diverges. The diagram below shows that the terms of the sequence grow without bound.

Since we conclude that the sequence diverges. The diagram below shows that the terms of the sequence grow without bound.

Does the sequence converge or diverge?

We compute Hence, the sequence converges to . The diagram below shows the terms of the sequence approaching .

We compute Hence, the sequence converges to . The diagram below shows the terms of the sequence approaching .

Does the sequence converge or diverge?

To compute the limit we use the fact that if a function, , has a limit, then the associated sequence has the same limit. Using L’Hopital’s rule twice, we have Hence, as well, and the sequence converges to .

To compute the limit we use the fact that if a function, , has a limit, then the associated sequence has the same limit. Using L’Hopital’s rule twice, we have Hence, as well, and the sequence converges to .

Show that the sequence is decreasing.

We will use the fact that if then . Since Taking reciprocals gives and hence the sequence is decreasing.

We will use the fact that if then . Since Taking reciprocals gives and hence the sequence is decreasing.

If the function associated with the sequence is an increasing (or decreasing)
function, then the sequence is increasing (or decreasing).

Show that the sequence is decreasing.

We will show that the associated function is an increasing function. To do this recall that if for , then is decreasing on the interval . The quotient rule gives which is negative if . Thus is decreasing on the interval and is a decreasing sequence.

We will show that the associated function is an increasing function. To do this recall that if for , then is decreasing on the interval . The quotient rule gives which is negative if . Thus is decreasing on the interval and is a decreasing sequence.

### Problems

You can type DNE or infinity into the answer box as needed.

(problem 2) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 3) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 4) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 5) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 6) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 7) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 8) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges

(problem 9) Find the limit of the sequence (if it exists), and then state whether the
sequence converges or diverges.

The sequence convergesdiverges

The sequence convergesdiverges