I have a question regarding infinite series. Is it true that \begin{equation} \frac{1}{n}\sum_{i=1}^n \cosh{\left(\frac{1}{3}\gamma^3x_i\right)}\underset{n\to\infty}\longrightarrow 1 \end{equation} if and only if \begin{equation}\frac{1}{n}\sum_{i=1}^n \vert x_i\vert \underset{n\to\infty}\longrightarrow 0,\end{equation} where $\gamma >0$ is a constant, and $\sup_i \vert x_i \vert < \infty$.
Any ideas (a proof?)??? Every help/hint is really much appreciated!
many thanks!