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10.8 Ratio Test for Convergence

Keywords

Ratio Test 比值判别法 • Absolute Convergence 绝对收敛

Factorials 阶乘 • Exponentials 指数函数 • Inconclusive 不确定/失效

The "Ultimate" Test for Series

The Ratio Test is one of the most versatile tools in the series toolkit. It is specifically designed to handle series containing factorials (\(n!\)) or exponentials (\(a^n\)), where other tests like the Integral Test or Comparison Test might be difficult to apply.

核心要点：The Ratio Test (LIM-7.A.11)

Let \(\sum a_n\) be a series with non-zero terms. Define the limit of the ratio of consecutive terms as:

\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

  1. If \(L < 1\): The series converges absolutely.

  2. If \(L > 1\) (or \(L = \infty\)): The series diverges.

  3. If \(L = 1\): The test is inconclusive. (The series could converge or diverge; you must use a different test, such as the \(p\)-series or Integral test).

When to Use the Ratio Test

The Ratio Test works by comparing the growth of the \((n+1)\)-th term to the \(n\)-th term.

  • Factorials: Terms like \(n!\) or \((2n)!\) are perfect for the Ratio Test because the ratio simplifies beautifully: \(\frac{(n+1)!}{n!} = n+1\).

  • Exponentials: Terms like \(3^n\) simplify as \(\frac{3^{n+1}}{3^n} = 3\).

Example: Evaluating \(\sum_{n=1}^{\infty} \frac{2^n}{n!}\)

  1. Set up the ratio:

    \[ \lim_{n \to \infty} \left| \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} \right| \]

  2. Simplify terms:

    \[ \lim_{n \to \infty} \left| \frac{2 \cdot 2^n}{(n+1) \cdot n!} \cdot \frac{n!}{2^n} \right| = \lim_{n \to \infty} \frac{2}{n+1} \]

  3. Evaluate the limit: The limit is 0. Since \(0 < 1\), the series converges absolutely.

Visualizing the Ratio

For a series to converge via the Ratio Test, the "multiplier" between terms must eventually settle down to a value less than 1. In the graph below, we plot the ratio \(R_n = |a_{n+1}/a_n|\) for the series \(\sum \frac{2^n}{n!}\).

Because the ratio values (points) fall and stay below the red boundary line (\(L=1\)), the series is guaranteed to converge.

Extension: The AP Exam Series Checklist

According to the official AP Exclusion Statement, you are only assessed on these six specific tests:

  1. nth Term Test (For divergence only)
  2. Integral Test
  3. Comparison Test (Direct)
  4. Limit Comparison Test
  5. Alternating Series Test
  6. Ratio Test

The Ratio Test is arguably the most crucial for the rest of Unit 10, as it is the primary method used to find the Radius of Convergence for Power Series.

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