Convergent and Divergent Sequences

Convergent Sequences

Convergent sequences have a finite limit.

Convergent Sequence

Limit = 0.

Convergent Sequence

Limit = 1.


Divergent Sequences

Divergent sequences do not have a finite limit.

Divergent Sequence

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 Convergent Sequence

a1= 3

a3= 1

a1000= 0.5012506253127.

a1000 000 = 0.5000012500006.

The limit is 0.5.

Convergent sequence.


2 Convergent Sequence

Convergent Sequence

a1= 0.5

a3= 0.6666

a1000= 0.999000999001

a1000 000 = 0.999999000001

The limit is 1.

Convergent sequence.