Answer
Convergent
Work Step by Step
Given: $a_n=\dfrac{n^3}{n^3+1}$
Here, we have
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}\dfrac{n^3}{n^3+1}$
This gives:
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}\dfrac{n^3/n^3}{n^3+1/n^3}$
or, $=\dfrac{\lim\limits_{n\to \infty}(1)}{\lim\limits_{n\to \infty} 1+\lim\limits_{n\to \infty}(1/n^3)}$
Thus, $a_n=1$
Hence, the sequence is convergent.