Limit of a Sequence

How to find the limit of a sequence and determine if the sequence is convergent or divergent: definition, 4 examples, and their solutions.

Convergent

Convergent

See the graph of an (the terms of a sequence).

As n goes to ∞,
an gets close to a constant value α.

Then you can write this like below:
limn → ∞ an = α.

The left side is read as
[the limit of an as n goes to infinity].

Just like this case,
if the limit value is a constant value (α),
then you can say that
an is convergent.

Divergent

If an is not convergent,
(does not get close a constant value)
then an is divergent.

Let's see three cases of divergent sequences.

Limit is ∞

See the graph of an.

As n goes to ∞,
an increases to infinity.

Then you can write this like below:
limn → ∞ an = ∞.

∞ is not a constant number:
it shows the state of increasing.
So an is divergent.

Limit is -∞

See the graph of an.

As n goes to ∞,
an decreases to -infinity.

Then you can write this like below:
limn → ∞ an = -∞.

-∞ is also not a constant number:
it shows the state of decreasing.
So an is divergent.

Oscillation

See the graph of an.

As n goes to ∞,
an does not go to one direction.

Then you can say that
an oscillates.

Example 1

Example

Solution

To see if an is convergent,
find the limit of an
as n → ∞.

an = 5 + 4/n2

So the limit of an is
the limit of 5 + 4/n2.

As n → ∞,
the constant 5 is still 5.

As n → ∞,
4/n2 goes to
4/∞2 = 4/∞ = 0.

If the denominator get bigger,
then the whole fraction gets smaller.
So 4/∞ = 0.

So the limit is 5 + 0.

5 + 0 = 5

The limit of an is 5.

This 5 is a constant number.

So an is convergent.

So an is convergent.

Example 2

Example

Solution

Find the limit of an
as n → ∞.

an = √n - 2

So the limit of an is
the limit of √n - 2.

As n → ∞,
n - 2 goes to
∞ - 2 = ∞.

Think of the graph of y = √n - 2.
As n → ∞,
the graph goes to ∞.

Graphing Square Root Functions

The limit of an is ∞.

∞ is not a constant number.

So an is divergent.

So an is divergent.

Example 3

Example

Solution

Find the limit of an
as n → ∞.

an = -n2 + 1

So the limit of an is
the limit of -n2 + 1.

As n → ∞,
-n2 goes to
-∞2 = -∞.

As n → ∞,
the constant +1 is still +1.

So the limit is -∞ + 1.

-∞ + 1 = -∞

The limit of an is -∞.

-∞ is not a constant number.

So an is divergent.

So an is divergent.

Example 4

Example

Solution

Find the first few terms of an.

a1 = -1
a2 = 1
a3 = -1
a4 = 1

The terms of an are -1, 1, -1, 1, ... .

an does not go to one direction.

So an shows oscillation.

an is oscillating.

So an is divergent.

So an is divergent.