Subtopic: Sequences and series – infinite series, tests for convergence
Question Statement:
The infinite series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if and only if:
Options:
(A) $p > 1$
(B) $p \ge 1$
(C) $p < 1$
(D) $p \le 1$
Answer Key:
A
Step-by-Step Solution & Explanation
By the integral test for convergence (the $p$-series test):
$\int_1^\infty \frac{1}{x^p} \, dx = \lim_{M \to \infty} \left[ \frac{x^{1-p}}{1 - p} \right]_1^M$
This improper integral converges if and only if $1 - p < 0 \iff p > 1$. When $p \le 1$, the integral and the corresponding series diverge.