10.3 The nth Term Test for Divergence
Keywords
nth Term Test 第n项判别法 • Divergence Test 发散测试
Inconclusive 不确定 • Necessary Condition 必要条件
The logic of the nth Term Test
If an infinite series is going to settle down to a finite sum, the terms you are adding must eventually become insignificantly small. If the terms \(a_n\) do not approach zero as \(n\) goes to infinity, the sum will continue to grow or oscillate forever.
Theorem: The nth Term Test for Divergence (LIM-7.A.5)
If \(\lim\limits_{n \to \infty} a_n \neq 0\), then the series \(\sum\limits_{n=1}^{\infty} a_n\) diverges.
Conversely, if \(\sum\limits_{n=1}^{\infty} a_n\) converges, then \(\lim\limits_{n \to \infty} a_n\) must equal \(0\).
The "Inconclusive" Trap
This is the most common mistake in Calculus BC. Students often assume that if the terms go to zero, the series converges. This is FALSE.
-
If \(\lim\limits_{n \to \infty} a_n \neq 0\): The series DIVERGES. (Certainty)
-
If \(\lim\limits_{n \to \infty} a_n = 0\): The test is INCONCLUSIVE. (The series might converge, or it might still diverge).
The Famous Counterexample: The Harmonic Series
Consider the series \(\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots\)
- Does the limit of the terms go to zero? Yes, \(\lim\limits_{n \to \infty} \frac{1}{n} = 0\).
- Does it converge? NO. The Harmonic series is a classic divergent series.
This proves that \(\lim a_n = 0\) is a necessary condition for convergence, but not a sufficient one.
Visualizing Divergence
In the graph below, we see the terms of \(a_n = \frac{n}{n+1}\). Notice that as \(n\) increases, the terms approach \(1\), not \(0\). Because we are essentially adding "\(1\)" over and over again at the end, the partial sums \(S_n\) will grow forever.
Extension: Divergence and the Limit Definition
Why does \(\lim a_n \neq 0\) guarantee divergence? Recall that \(a_n = S_n - S_{n-1}\). If the series converges to a sum \(S\), then as \(n \to \infty\):
This algebraic relationship is the formal proof that for a series to converge, its terms MUST approach zero.