Convergence of the integral $\int\limits_{1}^{\infty} \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+3}} \right) \, dx$ Would someone please help me prove that the integral
$$
\int\limits_{1}^{\infty} \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+3}} \right) \, dx
$$
is convergent?
Thank you.

Using $$\int x^m dx=\frac{x^{m+1}}{m+1}+C$$ for $m+1\ne0$
We have $$\int\left(\frac1{\sqrt x} -\frac1{\sqrt{x+3}} \right) \, dx=2(\sqrt x-\sqrt{x+3})+K$$
$$=2\frac{x-(x+3)}{\sqrt x+\sqrt{x+3}}+K=K-\frac6{\sqrt x+\sqrt{x+3}}$$
