# 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 a_{n} (the terms of a sequence).

As n goes to ∞,

a_{n} gets close to a constant value α.

Then you can write this like below:

lim_{n → ∞} a_{n} = α.

The left side is read as

[the limit of a_{n} as n goes to infinity].

Just like this case,

if the limit value is a constant value (α),

then you can say that

a_{n} is convergent.

## Divergent

If a_{n} is not convergent,

(does not get close a constant value)

then a_{n} is divergent.

Let's see three cases of divergent sequences.

### Limit is ∞

See the graph of a_{n}.

As n goes to ∞,

a_{n} increases to infinity.

Then you can write this like below:

lim_{n → ∞} a_{n} = ∞.

∞ is not a constant number:

it shows the state of increasing.

So a_{n} is divergent.

### Limit is -∞

See the graph of a_{n}.

As n goes to ∞,

a_{n} decreases to -infinity.

Then you can write this like below:

lim_{n → ∞} a_{n} = -∞.

-∞ is also not a constant number:

it shows the state of decreasing.

So a_{n} is divergent.

### Oscillation

See the graph of a_{n}.

As n goes to ∞,

a_{n} does not go to one direction.

Then you can say that

a_{n} oscillates.

## Example 1

### Example

### Solution

To see if a_{n} is convergent,

find the limit of a_{n}

as n → ∞.

a_{n} = 5 + 4/n^{2}

So the limit of a_{n} is

the limit of 5 + 4/n^{2}.

As n → ∞,

the constant 5 is still 5.

As n → ∞,

4/n^{2} 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 a_{n} is 5.

This 5 is a constant number.

So a_{n} is convergent.

So a_{n} is convergent.

## Example 2

### Example

### Solution

Find the limit of a_{n}

as n → ∞.

a_{n} = √n - 2

So the limit of a_{n} 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 ∞.

Square Root Function: Graph

The limit of a_{n} is ∞.

∞ is not a constant number.

So a_{n} is divergent.

So a_{n} is divergent.

## Example 3

### Example

### Solution

Find the limit of a_{n}

as n → ∞.

a_{n} = -n^{2} + 1

So the limit of a_{n} is

the limit of -n^{2} + 1.

As n → ∞,

-n^{2} goes to

-∞^{2} = -∞.

As n → ∞,

the constant +1 is still +1.

So the limit is -∞ + 1.

-∞ + 1 = -∞

The limit of a_{n} is -∞.

-∞ is not a constant number.

So a_{n} is divergent.

So a_{n} is divergent.

## Example 4

### Example

### Solution

Find the first few terms of a_{n}.

a_{1} = -1

a_{2} = 1

a_{3} = -1

a_{4} = 1

The terms of a_{n} are -1, 1, -1, 1, ... .

a_{n} does not go to one direction.

So a_{n} shows oscillation.

a_{n} is oscillating.

So a_{n} is divergent.

So a_{n} is divergent.