**Given:**

The sequence is
an=2+(−1)nn
.

**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.

**Theorem used:**

If
limn→∞|an|=0
, then
limn→∞an=0
. (1)

**Calculation:**

Check whether the sequence is increasing, decreasing or not monotonic.

The first five terms of the sequence is as follows:

{2+(−1)nn}={2+(−1)11,2+(−1)22,2+(−1)33,2+(−1)44,2+(−1)55...} ={2−1,2+12,2−13,2+14,2−15,...}

={1,52,53,94,95...}
(2)

Since
a1<a2
but
a2>a3
, the sequence is neither increasing nor decreasing.

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

Check whether the sequence is bounded or not.

From equation (2), the sequence is bounded as
1≤an≤52 for all n
.

Another Method:

Obtain the limit of the sequence (the value of the term
an
as *n* tends to infinity).

limn→∞an=limn→∞(2+(−1)nn)=limn→∞(2)+limn→∞((−1)nn)
.

=2+limn→∞((−1)nn)
(2)

The absolute value of
(−1)nn
is,

|(−1)nn|={|(1)n|if n is even|(−1)n|if n is odd=1n

The absolute value of
(−1)nn
when
n→∞
is obtained as follows.

limn→∞|(−1)nn|=limn→∞(1n)=1∞=0

Since
limn→∞|(−1)nn|=0
and by Theorem (1), the value of
limn→∞(−1)nn=0
.

Substitute 0 for
limn→∞(−1)nn
in equation (1),

limn→∞an=2+0=2

That is, the sequence
an=2+(−1)nn
is convergent to the limit 2.

Note that, if the sequence is convergent, then it is bounded.

Thus, the sequence is bounded.

Therefore, it can be concluded that the sequence is bounded but not monotonic.