## Calculus (3rd Edition)

Given $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{3 / 2}+1}}$$ Compare with $\sum \frac{1}{n^{1/4}}$, a divergent series ($p\lt1$): \begin{align*} \lim_{n\to \infty } \frac{n}{\sqrt{n^{3 / 2}+1}}.n^{1/4}&=\lim_{n\to \infty } \frac{1}{\sqrt{1+1/n^{3/2}}}\\ &=1 \end{align*} Hence, $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{3 / 2}+1}}$ also diverges.