Solution
= Solution
Compare the positive terms with those of the <harmonic series>:
$$
\frac{(n^2+2n)/(n^3+3n^2+1)}{1/n}=\frac{n^3+2n^2}{n^3+3n^2+1}\longrightarrow1.
$$
By the <limit comparison test>, the given <series> has the same convergence behaviour as $\sum1/n$, which diverges. Thus \b[the <series> diverges to $+\infty$].