Mathematics

The $n$th-term test (divergence test) says a series can only converge if its terms shrink to zero. Here, you need to find when $\frac{n^p}{n^p+1}$ does not approach zero as $n \to \infty$. The key insight is comparing the growth rates of numerator and denominator: when $p$ is large enough, $n^p$ dominates the "+1" in the denominator, so the fraction approaches a nonzero limit. Determine the threshold value of $p$ where this happens.

The $n$th-term test (divergence test) says a series can only converge if its terms shrink to zero. Here, you need to find when $\frac{n^p}{n^p+1}$ does not approach zero as $n \to \infty$. The key insight is comparing the growth rates of numerator and denominator: when $p$ is large enough, $n^p$ dominates the "+1" in the denominator, so the fraction approaches a nonzero limit. Determine the threshold value of $p$ where this happens.