Convergence of the following series as $\alpha \in \mathbb{R}$
$$\sum_{n=1}^{\infty}\left(\left(\sqrt{n}\right)\left(n^{\alpha\cos(\frac{1}{n})}-n-\cos\left(\frac{1}{n}\right)\right)\right)^{-1}$$
As $n \to +\infty$ we have $a_n\sim \left(n^{\frac{1}{2}}\left(n^{\alpha}-n-1\right)\right)^{-1} = \left(n^{\frac{2\alpha+1}{2}}-n^\frac{3}{2}-n^\frac{1}{2}\right)^{-1}$
Hence, as $0<\alpha<1$ we have $a_n =\mathcal{O}\left(\frac{1}{n^\frac{3}{2}}\right)$, as $n \to +\infty$, that converges.
As $\alpha>1$ we have $a_n=\mathcal{O}\left(\frac{1}{n^\frac{2\alpha+1} {2}}\right)$, as $n \to +\infty$, which converges in turn.
Is it right or I got rid of too much informations in the asymptotic expansion?
We have that
$$\cos\left(\frac{1}{n}\right)=1+O\left(\frac1{n^2}\right)$$
$$n^{\alpha\cos(\frac{1}{n})}=e^{\alpha\cos(\frac{1}{n})\log n}=e^{\alpha\log n+O\left(\frac{\log n}{n^2}\right)}=n^{\alpha}\left(1+O\left(\frac{\log n}{n^2}\right)\right)=n^{\alpha}+O\left(\frac{\log n}{n^{2-\alpha}}\right)$$
$$n^{\alpha\cos(\frac{1}{n})}-n-\cos\left(\frac{1}{n}\right)=n^{\alpha}+O\left(\frac{\log n}{n^{2-\alpha}}\right)-n-1+O\left(\frac1{n^2}\right)$$
and therefore
$$\left[\sqrt{n}\left(n^{\alpha\cos(\frac{1}{n})}-n-\cos\left(\frac{1}{n}\right)\right)\right]^{-1} \sim-\frac1{n\sqrt n}$$
$$\left[\sqrt{n}\left(n^{\alpha\cos(\frac{1}{n})}-n-\cos\left(\frac{1}{n}\right)\right)\right]^{-1} \sim-\frac1{\sqrt n}$$
$$\left[\sqrt{n}\left(n^{\alpha\cos(\frac{1}{n})}-n-\cos\left(\frac{1}{n}\right)\right)\right]^{-1} \sim \frac1{n^{\alpha+\frac12}}$$
and the given series converges if and only if $\alpha \neq 1$.