# Convergent and Divergent Sequences

#### Convergent Sequences

Convergent sequences have a finite limit.

Limit = 0.

Limit = 1.

#### Divergent Sequences

Divergent sequences do not have a finite limit.

Limit = ∞.

#### Oscillating Sequences

Oscillating sequences are not convergent or divergent. Their terms alternate from upper to lower or vice versa.

1, 0, 3, 0, 5, 0, 7, ...

#### Alternating Sequences

Alternating sequences change the signs of its terms. They can be:

#### Convergent

1, −1, 0.5, −0.5, 0.25, −0.25, 0.125, −0.125,..

The even and odd terms have limit 0.

#### Divergent

1, 1, 2, 4, 3, 9, 4, 16, 5, 25, ...

The even and odd terms have limit +∞.

#### Oscillating

−1, 2, −3, 4 ,−5, ..., (−1)^{n } n

#### Examples

Study the following sequences and determine their type:

1

a_{1}= 3

a_{3}= 1

a_{1000}= 0.5012506253127.

a_{1000 000 }= 0.5000012500006.

The limit is 0.5.

**Convergent sequence.**

2

a_{1}= 0.5

a_{3}= 0.6666

a_{1000}= 0.999000999001

a_{1000 000 }= 0.999999000001

The limit is 1.

Convergent sequence.