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. Check out online math resources for more practice.

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.

Frequently Asked Questions

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Is the divergence test the same as the nth term test?

Yes. The divergence test and the nth term test are two names for the same idea: look at the limit of the terms aₙ as n → ∞. If that limit is not 0, or does not exist, the series diverges. Some textbooks also call it the nth term test for divergence, which is a good reminder that it can only prove divergence, never convergence.

What is the nth term test for divergence?

The nth term test for divergence says that if lim aₙ as n → ∞ is not 0, the series ∑aₙ diverges. The reason is simple: if the terms do not shrink toward zero, you keep adding pieces of noticeable size, so the sum cannot settle on a finite value. It is usually the first check to try on any series because it is quick.

What is an example of the divergence test?

A classic example is ∑ n/(n + 1). As n grows, n/(n + 1) gets closer and closer to 1, so the limit of the terms is 1, not 0. By the divergence test, the series diverges. Compare that with ∑ 1/n, where the terms do go to 0; there the test says nothing, even though that series also diverges.

What are the rules of the divergence test?

The divergence test has two rules. If lim aₙ ≠ 0 (or the limit does not exist), the series diverges. If lim aₙ = 0, the test is inconclusive, and you need another test, such as the integral, comparison or ratio test. The most common mistake is concluding that a series converges just because its terms approach 0, which the harmonic series ∑ 1/n shows is false.

Can the nth term test be used for alternating series?

Yes, the nth term test works for any series, including alternating ones. For example, in ∑ (−1)ⁿ · n/(n + 1) the terms swing between values near −1 and 1, so they do not approach 0 and the series diverges. If the terms of an alternating series do approach 0, use the alternating series test next to check for convergence.