Question

The series \( \sum_{n=1}^{\infty} \frac{1}{n} \)

• A. converges to 0.
• B. converges to 1.
• C. converges to both 0 and 1.
• D. does not converge.

Hint:

The harmonic series \( \sum \frac{1}{n} \) is the most famous example of a divergent series whose terms approach zero. Do not be fooled by the fact that the terms get smaller and smaller. The sum still grows without bound, just very slowly.

Answer 1

The Correct Option is D

Step 1: Introduction:
The harmonic series, ∑ 1/n = 1 + 1/2 + 1/3 + 1/4 + … , is the subject of this analysis. The objective is to ascertain whether this infinite series converges to a finite value or diverges.
Step 2: Method:
The p-series test provides a direct approach for convergence/divergence determination.
The p-series test: A series of the form ∑ 1/n^p
- converges if p > 1.
- diverges if p ≤ 1.
Step 3: Application of the Method:
Using the p-series test:
The harmonic series is a p-series with p = 1.
Based on the p-series test, since p = 1 (and thus p ≤ 1), the series diverges.
This implies the sum of the series terms increases indefinitely, without approaching a finite limit.
Step 4: Conclusion:
The harmonic series does not converge. Consequently, the series diverges.