Answer
The series diverges by the nth term test.
Work Step by Step
$\Sigma_{n=1}^{\infty} \frac{n}{\sqrt (n^2+1)}$
Let's use the nth term test:
If $\lim\limits_{n \to \infty}a_n\ne0$, then the series diverges.
$\lim\limits_{n \to \infty}\frac{n}{\sqrt (n^2+1)}\times\frac{\frac{1}{n}}{\frac{1}{\sqrt n^2}}$
$\lim\limits_{n \to \infty}\frac{1}{\sqrt (1+\frac{1}{n^2})}=1$
Since $1\ne0$, the series diverges by the nth term test.
