Whether sequence is increasing, decreasing, not monotonic and bounded or not.

The sequence is bounded but not monotonic.

Given:

The sequence is an=cosn .

Definition used:

(1) The sequence {an} is increasing if an<an+1 for all n≥1 . That is, a1<a2<a3<... .

(2) The sequence {an} is decreasing and an>an+1 for all n≥1 . That is, a1>a2>a3>... .

(3) If the sequence is either increasing or decreasing, then the sequence is called monotonic; otherwise it is not monotonic.

(4) If {an} is the sequence with m≤an≤M for m,M∈ℕ , then the sequence is bounded.

Calculation:

The first five terms of the sequence is as follows.

{cosn}={cos1,cos2,cos3,cos4,cos5,...}   ={0.54,−0.42,−0.99,−0.65,0.28,...}

Here, a1>a2 and a2>a3 but a3<a4 and a4<a5 .

Thus, it can be concluded that the sequence is neither increasing nor decreasing.

Therefore, by definition (3), the sequence is not monotonic.

Since −1≤cosn≤1 and by definition (4), the sequence is bounded.

Hence, the sequence is bounded but not monotonic.