Answer
Converges Absolutely
Work Step by Step
Let $a_n=\dfrac{1}{\sqrt {n(n+1)(n+2)}}$
Applying comparison, we have
$\Sigma_{n=1}^\infty \dfrac{1}{\sqrt {n(n+1)(n+2)}} \lt \Sigma_{n=1}^\infty \dfrac{1}{\sqrt {(n)(n)(n)}}=\Sigma_{n=1}^\infty \dfrac{1}{{\sqrt [3] {n}}}$
$\implies a_n=\Sigma_{n=1}^\infty \dfrac{1}{{\sqrt [3] {n}}}=\Sigma_{n=1}^\infty \dfrac{1}{n^{(3/2)}}$
Here, the series $\Sigma_{n=1}^\infty \dfrac{1}{n^{3/2}}$ shows a convergent p-series with $p=\dfrac{3}{2} \gt 1$ .
Thus, the series Converges Absolutely by the comparison test.
