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10.7 Alternating Series Test for Convergence

Keywords

Alternating Series 交错级数 • Alternating Series Test (AST) 交错级数判别法

Decreasing Magnitude 递减幅值 • Conditional Convergence 条件收敛

What is an Alternating Series?

An alternating series is a series whose terms alternate between positive and negative signs. It typically takes one of these two forms:

\[ \sum_{n=1}^{\infty} (-1)^n a_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^{n+1} a_n \]

where \(a_n > 0\) for all \(n\).

The Alternating Series Test (AST)

The AST is a relatively "easy" test because it only requires two conditions to be met for the series to converge.

核心要点：The Alternating Series Test (LIM-7.A.10)

An alternating series \(\sum (-1)^n a_n\) converges if it satisfies the following two conditions:

  1. Decreasing Magnitude: The terms (ignoring the sign) must be non-increasing. That is, \(a_{n+1} \le a_n\) for all \(n\) beyond some point.

  2. Limit is Zero: The limit of the terms must be zero: \(\lim\limits_{n \to \infty} a_n = 0\).

If both conditions are met, the series is guaranteed to converge.

Visualizing Alternating Convergence

In an alternating series, the partial sums \(S_n\) "bounce" back and forth. However, if the terms are getting smaller and heading toward zero, the "bounce" becomes smaller and smaller, eventually trapping the sum at a single finite value.

The points above represent the partial sums of the Alternating Harmonic Series \(\sum \frac{(-1)^{n+1}}{n}\). Notice how they "squeeze" toward the limit (red line).

Important Distinctions

  1. Failure of Condition 2: If \(\lim\limits_{n \to \infty} a_n \neq 0\), the series diverges by the nth Term Test.

  2. Failure of Condition 1: If the terms don't decrease, the series might still diverge or oscillate, and the AST cannot be used to prove convergence.

Extension: Absolute vs. Conditional Convergence

The Alternating Harmonic Series \(\sum \frac{(-1)^{n+1}}{n}\) is a special case.

  • The series itself converges (by AST).
  • If you take the absolute value of the terms, you get the Harmonic Series \(\sum \frac{1}{n}\), which diverges.

When a series converges as an alternating series but diverges when all terms are made positive, we call it Conditionally Convergent. If it converges even with absolute values, it is Absolutely Convergent.

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