Determine whether the sequence converges or diverges.If it converges, find the limit.
$a_{n}=\frac{3 \sqrt{n}}{\sqrt{n}+2}$
To find: Whether the sequence converges or diverges and to obtain the limit if the sequence converges.
The sequence converges and its limit is 3.
Given:
The sequence is an=3nn+2.
Definition used:
If an is sequence and limn→∞an exists then the sequence an is convergent otherwise it is divergent.
Calculation:
Obtain the limit of the sequence to investigate whether the sequence converges or diverges.
Compute limn→∞an=limn→∞3nn+2.
Divide the numerator and the denominator by n.
limn→∞an=limn→∞3nnn+2n=limn→∞3nn+2n
Redefine the terms of the numerator and the denominator as follows:
limn→∞an=limn→∞31+2n=limn→∞(3)limn→∞(1)+limn→∞2n
Apply the property limn→∞(kna)=0 and simplify the numerator and the denominator.
limn→∞an=31+2∞=31+0=31=3
Since limn→∞an exist and by using the definition of the sequence, it can be concluded that the sequence converges to the limit 3.
Therefore, the sequence converges to the limit 3.