Mathematics

This is an alternating series, so you need to check two things: whether the terms decrease to zero, and whether absolute convergence holds. The key insight is that the exponential $e^n$ in the denominator dominates any polynomial $n^p$ as $n \to \infty$, which handles the limit condition easily. The real question is whether $\sum \frac{n^p}{e^n}$ converges absolutely—and this boils down to comparing growth rates: does the exponential decay overpower the polynomial growth for your value of $p$?

This is an alternating series, so you need to check two things: whether the terms decrease to zero, and whether absolute convergence holds. The key insight is that the exponential $e^n$ in the denominator dominates any polynomial $n^p$ as $n \to \infty$, which handles the limit condition easily. The real question is whether $\sum \frac{n^p}{e^n}$ converges absolutely—and this boils down to comparing growth rates: does the exponential decay overpower the polynomial growth for your value of $p$?