Question

Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) 

$$ \begin{gathered} a_{n}=\ln \left(2 n^{2}+6\right)-\ln \left(n^{2}+6\right) \\ \lim _{n \rightarrow \infty} a_{n}= \end{gathered} $$

Answer 1

The sequence is convergent.

Explanation:

Note that the given sequence is \(\left(a_{n}\right)\) where \(a_{n}=\log \left(\frac{2 n^{2}+6}{n^{2}+6}\right)\).

Therefore \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \log \left(\frac{2 n^{2}+6}{n^{2}+6}\right)\)

\(=\lim _{n \rightarrow \infty} \log \left(\frac{2+6 / n^{2}}{1+6 / n^{2}}\right)=\log (2)\)

The sequence is convergent.