Answer
Convergent
Work Step by Step
Given: $a_n=\dfrac{n(n+1)}{2n^2}$
Here, we have
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}$
This gives:
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}\dfrac{n^2+n/n^2}{2n^2/n^2}$
or, $=\dfrac{\lim\limits_{n\to \infty}(1)+\lim\limits_{n\to \infty} (1/n)}{\lim\limits_{n\to \infty} (2)}$
Thus, $a_n=\dfrac{1}{2}$
Hence, the sequence is convergent.