Answer
Absolutely Convergent
Work Step by Step
Let us apply the Ratio Test to the given series.
$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} \dfrac{1+e^{-2n}}{e+e^{-(2n-1)}} \\=\lim\limits_{n \to \infty} \dfrac{1+0}{e+0} \\=\dfrac{1}{e} \lt 1$
This implies that the series is Absolutely Convergent by the ratio test.
