calculus - Does the following infinite series converge?? (Leibnitz)

$$
\sum_{n=1}^{\infty} (-1)^n \left( \frac{1}{\sqrt{4n+1}} - \frac{1}{\sqrt{5n+1}}\right)

$$

Will someone please help me validate my way?

After moving to a common denominator, and multiplying by the "conjugate", we obtain the series:
$$
\sum_{n=1}^{\infty} (-1)^n \left( \frac{n}{ \sqrt{5n+1} \sqrt{4n+1} (\sqrt{4n+1}+\sqrt{5n+1}) } \right)
$$

as for the absolute convergence: we can use limit comparison test with $\frac{1}{n^{0.5}}$ to obtain the series diverges. As for convergence- this is a Leibnitz series, so it converges.

Am I right?

Thanks