Answer
The series diverges
Work Step by Step
$if\lim _{n\rightarrow \infty }a_{n}\neq 0\Rightarrow \sum ^{n}_{1}a_{n}$ diverges
$\lim \dfrac {n}{\sqrt {n^{2}+1}}=\dfrac {n/n}{\sqrt {n^{2}+1}/n}=\dfrac {1}{\sqrt {1+\dfrac {1}{n^{2}}}}=1\neq 0$
So the series diverges
