Answer
Absolutely Convergent
Work Step by Step
We need to apply the Ratio Test to the series.
$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} \dfrac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}}$
or, $=\lim\limits_{n \to \infty} \dfrac{1+e^{-2n}}{e+e^{-2n-1}}$
or, $=\lim\limits_{n \to \infty} \dfrac{1+0}{e+0}$
So, $=\dfrac{1}{e} \lt 1$
Thus, the series is Absolutely Convergent by the ratio test.