## Applications to the convergence of series

$$\displaystyle \sum_{n=1}^{\infty}\sin\left(\frac{1}{n}\right)$$
$$\displaystyle \sum_{n=1}^{\infty}\left(1-\cos\left(\frac{1}{n}\right)\right)n^{\alpha}$$ depending on the parameter $$\alpha\in\mathbb R$$.