Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
Whether the sequence is increasing, decreasing, not monotonic and bounded or not.
The sequence is monotonic but not bounded.
(1) The sequence
is increasing if
. That is,
(2) The sequence
is decreasing and
. That is,
(3) If the sequence is either increasing or decreasing, then the sequence is called monotonic; otherwise it is not monotonic.
is the sequence with
, then the sequence is bounded.
The sequence is
Check whether the sequence is increasing, decreasing or not monotonic.
The graph of the sequence
is shown below in Figure 1.
Here, it is observed that the sequence is increasing on
Since the sequence increases and by definition (3), the sequence is monotonic.
Check whether the sequence is bounded or not.
Obtain the first term of the sequence by substituting 1 for n in equation (1).
Thus, the first term of the sequence is
Therefore, the sequence is bounded below by 1.
1<an for all n≥1
Obtain the limit of the sequence (the value of the term
as n tends to infinity).
The limit of the sequence does not exist, the sequence is divergent. So, the sequence is not bounded above.
Thus, the sequence is not bounded.
Therefore, it can be concluded that the sequence is monotonic but not bounded.