How to study if an alternating series with squareroot n minus n is absolute convergent or simply convergent?

So, I am supposed to tell if the following series is absolute convergent or simply convergent: $$\sum(-1)^n\left(\sqrt{n^2+1}-n\right)$$

To study ther absolute convergence I know that I need to study the convergence of the following series: $$\sum \left| (-1)^n\left(\sqrt{n^2+1}-n\right) \right|= \sum\left(\sqrt{n^2+1}-n\right)$$

But I'm stuck on how to study such series.

Also, if it's not absolute convergent, how do I study the convergence of the alternating series?

Thanks in advance.

Solution 1:

We study the absolute convergence. We know that: $$\sqrt{1+n^2}-n=\frac{1}{n}\left(\sqrt{1+\frac{1}{n^2}}-n\right)=n\cdot \left(\sqrt{1+\frac{1}{n^2}}-1\right)\sim n\cdot \frac{1}{n^2}=\frac{1}{n}\,\,\, n\to +\infty$$ So, the series is not absolutely convergent by the asymptotic criteria and by comparison test with $\sum_{n=1}^{+\infty}\frac{1}{n}$ that diverges.

The series is convergent pointwise. Namely, $\sqrt{n^2+1}-n$ is decreasing, infinitesimal because: $$\lim_{n\to +\infty}\sqrt{n^2+1}-n=0$$ And always non-negative because $n^2+1>n^2\,\, \forall n$.

So, the seriesis pointwise convergent by the Leibniz criteria.