help? sequences?

Question

Answer 1

(1)

Step 1:

The sequence is .

.

Apply infinite limit on each side.

.

Formula : If  for ,  the sequence is divergent.

.

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

Solution :

The sequence is divergent and value of the limit is .

Answer 2

(2)

Step 1:

The sequence is .

.

Apply infinite limit on each side.

Formula : If  for ,  the sequence is divergent.

Formula :  If for , the sequence is divergent.

.

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

Solution :

The sequence is divergent.

Answer 3

(3)

Step 1:

The sequence is .

.

Apply infinite limit on each side.

.

First calculate the absolute value of the sequence.

Formula : If  for ,  the sequence is divergent.

.

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

Solution :

The sequence is divergent.

Answer 4

(4)

Step 1:

The sequence is .

.

Apply infinite limit on each side.

.

First calculate the absolute value of the sequence.

Formula :  If for , 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.