(1)

Step 1:

The sequence is .

Find the limit of the sequence if convegent.

.

Apply on each side.

.

Theorem :  for .

.

Hence the sequence is convergent and value of the limit is .

Solution : \ \

.

(2)

Step 1:

The sequence is .

Find the limit of the sequence if convegent.

.

Apply on each side.

.

Theorem :  for .

Theorem :  for , hence the sequence is divergent.

.

Hence the sequence is divergent and value of the limit is .

Solution : \ \

.

(3)

Step 1:

The sequence is .

Find the limit of the sequence if convegent.

.

Apply on each side.

.

First calculate the absolute value of the sequence.

Theorem :  for .

.

Hence the sequence is convergent and value of the limit is .

Solution : \ \

.

(4)

Step 1:

The sequence is .

Find the limit of the sequence if convegent.

.

Apply on each side.

.

First calculate the absolute value of the sequence.

Theorem :  for , hence the sequence is divergent.

does not exist as it oscillates between and .

Hence the sequence is divergent and limit of the sequence does not exist.

Solution : \ \

The sequence is divergent and limit of the sequence does not exist.