Determine whether the sequence converges or diverges.If it converges, find the limit.

$a_{n}=3^{n} 7^{-n}$

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

The sequence is converges and its limit is 0.

Given:

The sequence is an=3n7−n .

Definition used:

If an is sequence and limn→∞an is exists then the sequence an is convergent otherwise it is divergent.

Result used:

The sequence {rn} is converges to zero when −1<r<1 .

That is, limn→∞rn=0 if −1<r<1 .

Calculation:

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

limn→∞an=limn→∞(3n7−n) =limn→∞(3n7n) =limn→∞(37)n

Since 37<1 and by using the result, limn→∞an=0 .

Since limn→∞an is exist and by using the definition of the sequence, it can be concluded that the sequence is converges to the limit zero.

Therefore, the sequence is converges to the limit 0 when r=37 .