Answer
Converges
Work Step by Step
Given $$\sum_{n=25}^{\infty} \frac{n^2}{(n^3+9)^{5/2}}$$
Since $f(n)= \frac{n^2}{(n^3+9)^{5/2}},\ \ f'(n) = \frac{n\left(-11n^3+36\right)}{2\left(n^3+9\right)^{\frac{7}{2}}}$, is positive, decreasing and continuous for $n\geq25$, then by using the integral test
\begin{aligned} \int_{25}^{\infty} f(n) dn &=\lim _{t \rightarrow \infty} \int_{25}^{t} \frac{n^2}{(n^3+9)^{5/2}}d n \\
&=\left.\lim _{t \rightarrow \infty}\frac{-2}{9} (n^3+9)^{-3/2}\right|_{25} ^{t} \\
&= \lim _{t \rightarrow \infty} \frac{ -2}{9(n^3+9)^{3/2}}+\lim _{t \rightarrow \infty} \frac{ 2}{9(25^3+9)^{3/2}} \\ &=\frac{\sqrt{15634}}{1099898802}\end{aligned}
Thus the series converges.
