A square root whose difference tends to 1/2

Rationalising.
$$\sqrt{n^2+n+1}-n=\frac{(n^2+n+1)-n^2}{\sqrt{n^2+n+1}+n}=\frac{n+1}{\sqrt{n^2+n+1}+n}.$$
Dividing numerator and denominator by $n$,
$$=\frac{1+\frac1n}{\sqrt{1+\frac1n+\frac1{n^2}}+1}\longrightarrow\frac12.$$

Conclusion. The general term tends to $\frac12\ne0$, so the series diverges (divergence test).