$\lim_{n \to \infty} \frac{a_n}{b_n} = 1$; $\sum b_n$ diverges, so original series diverges

$\lim_{n \to \infty} \frac{a_n}{b_n} = 0$; series converges

$\lfloor n^2 + 2n \rfloor = n^2 + 2n$ for all $n$, so series reduces to $\sum \frac{1}{1+2/n}$, which converges

Test is inconclusive; floor function makes comparison invalid