Sequences Problems

Solutions

Interpret the Following Sequences and Describe Them.

1an = 1, 2, 3, 4, 5, ... n.

2an = -1, −2,−3, −4, −5, ... −n.

3an = 2, 3/2, 4/3, 5/4, ..., n+1/n.

4an= 2, −4, 8, −16, 32, ..., (−1)n−1 2n.

5Monotonically Decreasing Sequence

6Non-Monotonous Sequence7Non-Monotonous Sequence8


1

an = 1, 2, 3, 4, 5, ... n

It is increasing.

It is bounded below.

1 is the infimum.

It is not bounded above.

It is divergent.


2

an = −1, −2,−3, −4, −5, ... −n

It is decreasing.

It is bounded above.

−1 is the supremum.

It is not bounded below.

It is divergent.


3

an = 2, 3/2, 4/3, 5/4, ..., n+1/n

It is decreasing.

It is bounded above

2 is the supremum.

It is bounded below.

1 is the infimum.

It is convergent and the limit is 1.


4

an= 2, −4, 8, −16, 32, ..., (−1)n−1 2n

It is not monotonous

It is not bounded.

It is neither convergent nor divergent.


5

Monotonically Decreasing

Monotone

3, 4/3, 1, 6/7,...

It is strictly monotonically decreasing.

Limit

a1= 3

a3= 1

a1 000= 0.5012506253127

a1 000 000 = 0.5000012500006

The limit is 0.5.

It is a convergent sequence.

Bounded

As the sequence is decreasing, 3 is an upper bound and the supremum.

0.5 is a lower bound and the infimum.

Thus, the sequence is bounded.

0.5 < a n ≤ 3.


6

Non-Monotonous Sequence

2, − 4, 8, −16, ...

It is not monotonous.

It is neither convergent nor divergent.

It is not bounded.


7

Non-Monotonous Sequence

It is not monotonous.

It is convergent, the limit is 0.

It is bounded above and 1 is the supremum.

It is bounded below and −1 is the infimum.

It is bounded.

−1 ≤a n ≤ 1.


8

Monotonically Increasing Sequence

Monotone

It is strictly monotonically increasing.

Limit

a1= 0.5.

a3= 0.6666 .

a1000= 0.999000999001.

a1000 000 = 0.999999000001.

The limit is 1.

It is a convergent sequence.

Bounded

It is bounded below and 1/2 is the infimum.

It is bounded above and 1 supremum.

The sequence is bounded.

0.5 ≤ a n < 1