Ratio and Root Tests: Chapter Study Guide

Learning Objectives

By the end of this study guide, you should be able to:

• Use the ratio test to determine the absolute convergence of a series involving factorials and exponential terms.
• Use the root test to determine the absolute convergence of a series where the $n-th term is raised to the n$-th power.
• Describe a general strategy for evaluating an unknown series and choosing the most efficient convergence test.
• Understand the limitations of the ratio and root tests, specifically identifying when they yield inconclusive results.

Key Terms & Glossary

• Absolute Convergence: A series $\sum a_n converges absolutely if the series of absolute values \sum |a_n|$ converges. Absolutely convergent series always converge.
• Ratio Test: A test that evaluates the limit of the absolute ratio of consecutive terms to determine convergence rates relative to a geometric series.
• Root Test: A test that evaluates the $n-th root of the absolute value of the n$-th term to determine convergence rates.
• Factorial ($n!$): The product of an integer and all the integers below it (e.g., $$4! = 4 \times 3 \times 2 \times 1$$). Factorials grow faster than exponential functions.
• Inconclusive Result: A scenario where a specific test provides no definitive answer about convergence or divergence, requiring a different test to be applied.

The "Big Idea"

The Ratio and Root Tests measure how quickly the terms of a series approach zero. Unlike the Comparison Test or Limit Comparison Test, they do not require you to find a secondary comparable series. Instead, they evaluate the series purely based on its own terms.

By calculating a limit (often denoted as $$\rho$), these tests essentially determine if the tail end of your series behaves like a convergent or divergent geometric series. They are the most powerful tools in your arsenal when dealing with power series, factorials, and functions raised to the n$-th power.

[!NOTE] Both tests evaluate absolute convergence. If the limit \rho < 1$, the series converges absolutely. If $\rho > 1$, the terms do not approach zero fast enough, and the series diverges. If $\rho = 1, the tests are "blind" to the behavior, and you must use a different test.

Formula / Concept Box

Hierarchical Outline

• 1. Introduction to Ratio & Root Tests
  • The purpose: Determining absolute convergence without a comparison series.
  • The connection to geometric series: Using $\rho$ as a common ratio analogue.
• 2. The Ratio Test
  • When to use: Best for series containing factorials ($n!) or constants raised to the n$-th power ($c^n$).
  • Simplification strategy: Cancel common factors between $a_{n+1}$ and $a_n$.
• 3. The Root Test
  • When to use: Best for series where the entire sequence formula is raised to the $n$-th power.
  • Algebraic strategy: The $n-th root cancels the n$-th power, simplifying the limit evaluation.
• 4. General Strategy for Convergence Testing
  • Step 1: Identify familiar forms ($p$-series, geometric).
  • Step 2: Check for alternating terms.
  • Step 3: Look for factorials (use Ratio) or large overarching powers (use Root).
  • Step 4: Fall back on Divergence, Comparison, or Integral tests if $\rho = 1$.

Visual Anchors

Convergence Test Decision Flowchart

The Limit Boundary ($\rho$)

Definition-Example Pairs

• Term: The Ratio Test

  • Definition: Taking the limit of the ratio of the next term over the current term to check if the series shrinks fast enough to converge.
  • Example: For $\sum \frac{1}{n!}$, $a_{n+1} = \frac{1}{(n+1)!}$. The ratio is $\frac{n!}{(n+1)!} = \frac{1}{n+1}$, which approaches 0 as $n \to \infty$. Since $0 < 1$, it converges.

• Term: The Root Test

  • Definition: Taking the $n$-th root of the $n$-th term to see if the exponential base of the sequence is less than 1.
  • Example: For $\sum \left( \frac{n}{2n+1} \right)^n$, taking the $n$-th root yields $\frac{n}{2n+1}$. As $n \to \infty$, this limit is $1/2$. Since $1/2 < 1$, it converges.

Worked Examples

Example 1: Applying the Ratio Test

Determine if the series $\sum_{n=1}^\infty \frac{10^n}{n!}$ converges or diverges.

Step 1: Set up the ratio limit $\rho$. $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{10^{n+1}}{(n+1)!}}{\frac{10^n}{n!}} \right|$

Step 2: Multiply by the reciprocal and group like terms. $\rho = \lim_{n \to \infty} \left( \frac{10^{n+1}}{10^n} \cdot \frac{n!}{(n+1)!} \right)$

Step 3: Simplify the exponents and the factorials. (Remember $(n+1)! = (n+1)n!$). $\rho = \lim_{n \to \infty} \left( 10 \cdot \frac{1}{n+1} \right)$

Step 4: Evaluate the limit. $\rho = 0$ Since $\rho = 0 < 1$, the series converges absolutely by the Ratio Test.

[!TIP] Whenever you see a constant raised to an $n$-th power competing against a factorial, the factorial always wins eventually. The Ratio test proves this mathematically by driving the limit to 0.

Example 2: Applying the Root Test

Determine if the series $\sum_{n=1}^\infty \left( \frac{3n+2}{5n-1} \right)^{2n}$ converges or diverges.

Step 1: Notice the entire sequence is raised to a power involving $n. Apply the Root Test limit \rho$. $\rho = \lim_{n \to \infty} \sqrt[n]{\left| a_n \right|} = \lim_{n \to \infty} \sqrt[n]{\left( \frac{3n+2}{5n-1} \right)^{2n}}$

Step 2: Simplify the root and power (the $n in the exponent cancels with the n$-th root). $\rho = \lim_{n \to \infty} \left( \frac{3n+2}{5n-1} \right)^2$

Step 3: Evaluate the limit of the inside function, then square it. As $n \to \infty$, $\frac{3n+2}{5n-1} \to \frac{3}{5}$. $\rho = \left( \frac{3}{5} \right)^2 = \frac{9}{25}$

Step 4: Conclude. Since $\rho = 0.36 < 1$, the series converges absolutely by the Root Test.

Checkpoint Questions

1. If applying the Ratio Test yields $\rho = 1$, what can you conclude about the series? What should you do next?
2. Why is the Ratio Test particularly well-suited for series containing factorials like $(2n)!$?
3. Calculate the limit $\rho$ for the series $\sum \frac{n^2}{2^n}$ using the Ratio Test. Does it converge?
4. Without calculating, why might the Root Test be a poor choice for the harmonic series $\sum \frac{1}{n}$?
5. True or False: If a series converges conditionally, the Ratio Test will always yield $\rho < 1$.

▶Click here to check your answers

1. The test is inconclusive. You must select a different test, such as the limit comparison test, alternating series test, or integral test.
2. Factorials expand nicely when calculating the ratio of the next term. $(n+1)!$ simply becomes $(n+1) \cdot n!$, allowing the factorials to cancel completely out of the expression.
3. Yes, it converges. The ratio simplifies to $\frac{1}{2} \lim_{n\to\infty} (\frac{n+1}{n})^2 = \frac{1}{2} (1) = \frac{1}{2} < 1$.
4. The $n$-th root of a polynomial expression like $1/n$ will evaluate to a limit of 1 (inconclusive), wasting your time.
5. False. The Ratio test checks for absolute convergence. For conditionally convergent series (like the alternating harmonic series), the Ratio test evaluates to $\rho = 1$ (inconclusive).