As you've seen from the title, I'm wondering whether the series $\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}$ converges or diverges? I'm struggling!
My first thought was to rewrite it like this to determine an alternating series: $$\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}=\sum_{n=2}^{\infty}\frac{(-1)^{n}}{(-1)^{n}+\sqrt{n}}$$
After seeing that $\frac{1}{(-1)^{n}+\sqrt{n}}$ isn't a decreasing sequence I can't use Leibniz's theorem (alternating series test). I would be thankful for some guidance on where to go next, I'm stuck!
As
$$\begin{aligned}\frac{1}{1+(-1)^{n}\sqrt{n}}&=\frac{(-1)^n}{\sqrt n}\frac{1}{1+(-1)^n/\sqrt n}\\ &= \frac{(-1)^n}{\sqrt n}\left(1-\frac{(-1)^n}{\sqrt n}+\frac{1}{n}+o\left(\frac{1}{n}\right)\right)\\ &= \frac{(-1)^n}{\sqrt n}-\frac{1}{n}+ \frac{(-1)^n}{n^{3/2}}+ o\left(\frac{1}{n^{3/2}}\right) \end{aligned}$$
the initial series diverges as the harmonic series diverges.