Series Behavior with the n-th Term Test: Divergence Test

The n-th Term Test is a fundamental method for determining the divergence of an infinite series. The test examines the behavior of the terms of the series as nn approaches infinity.

How It Works:

Given a series \(\sum a_n\), the n-th Term Test states:

• If \(\lim_{n \to \infty} a_n \neq 0\), the series diverges.
• If \(\lim_{n \to \infty} a_n = 0\), the test is inconclusive, meaning the series might converge or diverge. Further tests are needed.

Why It Works:

The intuition behind the test is simple: For a series to converge, its terms must approach zero as nn grows large. If the terms do not approach zero, the series cannot possibly sum to a finite value, and thus, it diverges. However, if the terms approach zero, we cannot conclude convergence, as some series with terms approaching zero still diverge (like the harmonic series).

Example of Divergence:

Consider the series \(\sum_{n=1}^\infty \frac{1}{n}\), the harmonic series.

• The terms are \(\frac{1}{n}\).
• \(\lim_{n \to \infty} \frac{1}{n} = 0\).

Since the terms approach zero, the n-th Term Test is inconclusive. However, we know that the harmonic series diverges because the sum grows without bound, even though the terms approach zero.

Example of Divergence (non-zero limit):

Consider the series \(\sum_{n=1}^\infty 1\).

• The terms are \(a_n = 1\).
• \(\lim_{n \to \infty} 1 = 1\).

Since the terms do not approach zero, the n-th Term Test tells us the series diverges.

Conclusion:

The n-th Term Test is a quick and effective method for determining divergence. If the limit of the terms does not approach zero, the series must diverge. However, if the terms approach zero, more tests are necessary to determine if the series converges.