How can I find if the series $$\sum a_k =\sum (k-\sqrt{k^2+4k+3})$$ is convergent or divergent?
Let $a_{k}=k-\sqrt{k^2+4k+3}$.
Then $$\lim_{k \rightarrow \infty}\bigg(k-\sqrt{k^2+4k+3}\bigg)\cdot \frac{\bigg(k+\sqrt{k^2+4k+3}\bigg)}{k+\sqrt{k^2+4k+3}}.$$
$$\lim_{k\rightarrow \infty}\frac{-4k-3}{k+\sqrt{k^2+4k+3}}=-4(\neq 0).$$
So the series $\sum a_{k}$ is divergent.
Is my process correct? If not, how do I solve it? Help me please