Is the series comparison criterion for convergence valid for absolute convergence?

Question

I'm studying the convergence and absolute convergence of the series of functions defined by the sequence of functions: \begin{equation*} f_n: \mathbb{R} \to \mathbb{R}, \end{equation*}

\begin{equation*} \phantom{1000}x \mapsto \sin\left(\dfrac{x}{n^2}\right). \end{equation*}

I prove that $\forall x \in \mathbb{R}$, $\forall n \in \mathbb{N}$: \begin{equation*} \left|\sin\left(\dfrac{x}{n^2}\right)\right| \le \dfrac{|x|}{n^2}, \end{equation*}

and $\sum _{n=1}^{\infty }\:\frac{\left|x\right|}{n^2}$ is absolut convergent, does it means that $\sum _{n=1}^{\infty }\:\sin(\frac{x}{n^2})$ is absolute convergent?

And if it is why can I apply this?

Answer 1

Yes, just apply Comparision test