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.

Use
$$\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+3}}=\frac{\sqrt{x+3}-\sqrt{x}}{\sqrt{x^2+3x}}=\frac{3}{(\sqrt{x+3}+\sqrt{x})\sqrt{x^2+3x}}$$
So, the integrand is positive and $\le \frac{3}{2x\sqrt{x}}$.
Here I use $\sqrt{x^2+3x}>x$ and $\sqrt{x+3}>\sqrt{x}$
So the integral converges by the comparison criterium.
