Question Bank • QB_EM_CALC_015 MCQ • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Sequences and series – infinite series, tests for convergence

Question Statement:

The $p$-series $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent 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 on $f(x) = \frac{1}{x^p}$: $\int_1^\infty \frac{1}{x^p} dx = \lim_{M \to \infty} \left[\frac{x^{1-p}}{1-p}\right]_1^M$ This improper integral converges if and only if $1 - p < 0 \implies p > 1$. For $p = 1$, it is the harmonic series $\sum \frac{1}{n}$ which diverges.
← Previous (QB_SUB_EM_0062) Next (QB_EM_095) →

Practice In Authentic CBT Exam Interface

Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.

Launch CBT Mock Test