Solution
= Solution
The PDF has the doubly exponential denominator $e^{e^n}$. For its positive summands, the successive-term ratio is
$$
\frac{a_{n+1}}{a_n}=\left(1+\frac1n\right)^3\exp\bigl(e^n-e^{n+1}\bigr)=\left(1+\frac1n\right)^3e^{-(e-1)e^n}\longrightarrow0.
$$
Thus the <ratio test> proves \b[absolute convergence]. Concretely, the tail is eventually dominated by a <geometric series> of ratio $1/2$; the polynomial numerator cannot offset the doubly exponential decay.