Question

Determine if the following sequence converges or diverges. If the sequence converges, find its limit.

Answer 1

\[\left\{ (4n^2)/5n^3-2 \right\}, n=1,2,3...\]

Answer 2

We have the following limit:
\[\lim_{n \rightarrow \infty}\frac{4n^2}{5n^3-2}=\frac{\infty}{\infty}\]We can see that we can apply L'Hôpital's Rule to solve this problem:
\[\lim_{n \rightarrow \infty}\frac{8n}{15n^2}=\frac{\infty}{\infty}\]\[\lim_{n \rightarrow \infty}\frac{8}{30n}=\frac{8}{\infty}=0\]Therefore:
\[\lim_{n \rightarrow \infty}\frac{4n^2}{5n^3-2}=0\]

Answer 3

excellent, thanks!