Answer
Convergent
Work Step by Step
The Comparison Test states that the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ is convergent if $p\gt 1$ and divergent if $p\leq 1$.
Given: $\Sigma _{n=3}^{\infty}\frac{{n+2}}{(n+1)^{3}}$
It can be re-written as:
$\Sigma _{n=3}^{\infty}\frac{{n+2}}{(n+1)^{3}}=\Sigma _{n=3}^{\infty}\frac{{n+1+1}}{(n+1)^{3}}$
$=\Sigma _{n=3}^{\infty}\frac{{n+1}}{(n+1)^{3}}+\Sigma _{n=3}^{\infty}\frac{{1}}{(n+1)^{3}}$
Adding and subtracting a finite number of terms from a series does not affect the convergence or divergence of the series.
The first series is convergent because a $p-$ series with $p=2$ is convergent.
The second series is convergent because a $p-$ series with $p=3$ is convergent.
The sum of two convergent series is convergent.
