Why is 1/n divergent?
Understand the Problem
The question asks about the divergence of the series represented by 1/n, specifically why it does not converge to a finite limit as n increases. This involves analyzing the behavior of the terms in the series as n approaches infinity.
Answer
The harmonic series ∑ 1/n diverges.
The final answer is that the harmonic series ∑ 1/n diverges because the sum of its partial sums cannot be bounded.
Answer for screen readers
The final answer is that the harmonic series ∑ 1/n diverges because the sum of its partial sums cannot be bounded.
More Information
The divergence of \sum 1/n is illustrated by the harmonic series, which grows without bound despite the terms continually decreasing. Unlike convergent series, where the sum approaches a finite value, the harmonic series increases indefinitely.
Tips
A common mistake is assuming any series with decreasing terms will converge. However, it's important to check if the series' terms decrease quickly enough.
Sources
- Why does the infinite series 1/n diverge? : r/askscience - Reddit - reddit.com
- Why does 1/n diverge? - Quora - quora.com
