Whether the sequence is converges or diverges and obtain the limit if the sequence is converges.

The sequence is diverges.

Given:

Consider the sequence as an=sinn .

Definition used:

If an is a sequence and limn→∞an is exists, then the sequence an is said to be convergent otherwise it is divergent.

Calculation:

Obtain the limit of the sequence to investigate whether the sequence is converges or diverges.

Compute limn→∞an=limn→∞sin(n) . (1)

Use the subsequence n=2mπ in equation (1).

limn→∞sin(n)=limm→∞(sin(2mπ))=limm→∞(0)=0

Use the subsequence n=(2m+12)π in equation (1).

limn→∞sin(n)=limm→∞(sin((2m+12)π))               =limm→∞(1)               =1

Here, it is observed that the term an oscillates between 0 and 1.

Since limn→∞an does not exist and by using the definition of the sequence, it can be concluded that the sequence is diverges.

Therefore, the sequence is diverges.