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 
a1= 3
a3= 1
a1000= 0.5012506253127.
a1000 000 = 0.5000012500006.
The limit is 0.5.
Convergent sequence.
2 
a1= 0.5
a3= 0.6666
a1000= 0.999000999001
a1000 000 = 0.999999000001
The limit is 1.
Convergent sequence.
