# Ratio Test for Convergence

> AP Calculus BC · Unit 10: Infinite Sequences and Series
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u10-ratio-test-for-convergence/

This module covers the ratio test for infinite series convergence, including core rules, handling inconclusive cases, application to finding radius of convergence for power series, and common exam traps.

**Prerequisites:** Computing limits of ratios of sequences; Definition of absolute and conditional convergence; [Basics of infinite series convergence](https://www.owlsprep.com/study/ap-calculus-bc-u10-overview/)

## Learning objectives

- Apply the ratio test to determine absolute convergence or divergence of infinite series
- Recognize and correctly handle inconclusive cases of the ratio test
- Use the ratio test to calculate the radius of convergence for power series
- Avoid common exam pitfalls when applying the ratio test

## Core Concepts of the Ratio Test

The Ratio Test (also called d'Alembert's Ratio Test) is a convergence test for infinite series $\sum_{n=1}^\infty a_n$, most useful when series terms contain factorials, exponential functions, or powers of $n$ that simplify cleanly when taking the ratio of consecutive terms. This topic is tested on both multiple-choice and free-response sections of the AP Calculus BC exam.

**Ratio Test** — A convergence test that compares the asymptotic growth rate of consecutive terms of an infinite series. Three outcomes: 1) If $L < 1$, the series converges absolutely; 2) If $L > 1$ (including $L = \infty$), the series diverges; 3) If $L = 1$, the test is inconclusive.

*Notation:* L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

*Example:* The limit $L$ behaves like the absolute value of the common ratio for geometric series.

**Worked example:** Determine whether the series $\sum_{n=1}^\infty \frac{3^n}{n!}$ converges or diverges.

1. Identify $a_n$ and $a_{n+1}$:

   $$a_n = \frac{3^n}{n!}, \quad a_{n+1} = \frac{3^{n+1}}{(n+1)!}$$
2. Set up the ratio of consecutive terms (all terms are positive, so absolute value can be dropped):

   $$\frac{a_{n+1}}{a_n} = \frac{3^{n+1}}{(n+1)!} \cdot \frac{n!}{3^n}$$
3. Simplify the expression:

   $$\frac{3^{n+1}}{3^n} = 3, \quad \frac{n!}{(n+1)!} = \frac{1}{n+1} \implies \frac{a_{n+1}}{a_n} = \frac{3}{n+1}$$
4. Compute the limit as $n \to \infty$:

   $$L = \lim_{n \to \infty} \frac{3}{n+1} = 0$$
5. Apply the ratio test rule: $0 < 1$, so the series converges absolutely.

> **tip**
>
> Always take the absolute value of the ratio before computing the limit. This ensures you correctly test for absolute convergence, even for alternating series and power series with negative $x$ values.

## Inconclusive Cases

When $L = 1$, the ratio test gives no information about convergence, and this is a common testing point on the AP exam. The ratio test only captures exponential growth rates of terms; when growth is polynomial, the limit $L$ will always equal 1, regardless of whether the series converges or diverges. Whenever you calculate $L=1$, you must switch to another appropriate test.

**Worked example:** The ratio test for the series $\sum_{n=1}^\infty \frac{n}{2n^2 + 1}$ gives $L=1$. Determine whether the series converges or diverges.

1. Confirm that $L=1$ means the ratio test is inconclusive, so we use the limit comparison test.
2. For large $n$, the end behavior of $a_n$ is approximately $\frac{n}{2n^2} = \frac{1}{2n}$, so we compare to the divergent series $\sum \frac{1}{n}$.
3. Compute the limit of the ratio of terms:

   $$\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{n/(2n^2 +1)}{1/n} = \lim_{n \to \infty} \frac{n^2}{2n^2 +1} = \frac{1}{2}$$
4. Since the limit is positive and finite, the original series diverges by the limit comparison test.

> **tip**
>
> If you get $L=1$ on the AP exam, never stop and conclude convergence or divergence. You must explicitly state the ratio test is inconclusive and use a second test to earn full credit.

## Applying the Ratio Test to Power Series

The most common high-stakes use of the ratio test on the AP Calculus BC exam is finding the radius and interval of convergence for a power series of the form $\sum_{n=0}^\infty c_n (x-a)^n$. Convergence of a power series depends on the value of $x$, and the ratio test is ideal here because the $(x-a)$ term simplifies cleanly when taking the ratio of consecutive terms.

1. Write $a_n = c_n (x-a)^n$, then set up the ratio $\left| \frac{a_{n+1}}{a_n} \right|$
2. Separate terms involving $x$ from terms involving $n$
3. Compute the limit $L(x) = |x-a| \cdot \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right|$
4. Set $L(x) < 1$ to find the radius of convergence $R = 1 / \left( \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| \right)$
5. Check the endpoints $x = a \pm R$ separately with another test

**Worked example:** Find the radius of convergence for the power series $\sum_{n=1}^\infty \frac{(3x)^n}{n 4^n}$.

1. Identify $a_n$ and $a_{n+1}$:

   $$a_n = \frac{(3x)^n}{n 4^n}, \quad a_{n+1} = \frac{(3x)^{n+1}}{(n+1) 4^{n+1}}$$
2. Set up and simplify the absolute value ratio:

   $$\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(3x)^{n+1}}{(n+1) 4^{n+1}} \cdot \frac{n 4^n}{(3x)^n} \right| = \frac{3|x|}{4} \cdot \frac{n}{n+1}$$
3. Compute the limit as $n \to \infty$:

   $$\lim_{n \to \infty} \frac{n}{n+1} = 1 \implies L = \frac{3|x|}{4}$$
4. Set $L < 1$ for convergence:

   $$\frac{3|x|}{4} < 1 \implies |x| < \frac{4}{3}$$
5. The radius of convergence is $R = \frac{4}{3}$.

> **tip**
>
> AP FRQ questions almost always require endpoint checks after finding the radius with the ratio test. Never skip checking endpoints if the question asks for the full interval of convergence.

## AP-Style Concept Check

**Check your understanding**

Test your understanding with this multiple-choice question:

1. Which of the following is the radius of convergence of the power series $\sum_{n=0}^\infty \frac{n! x^n}{4^n}$?

   - 0
   - 4
   - $\frac{1}{4}$
   - $\infty$

   *Answer:* 0

   *Why:* Correct. Applying the ratio test gives $L = \infty$ for any $x \neq 0$, so the series only converges at $x=0$, giving a radius of convergence of 0.

## Common pitfalls

- **Wrong:** Reversing the ratio order, computing $\left| \frac{a_n}{a_{n+1}} \right|$ instead of $\left| \frac{a_{n+1}}{a_n} \right|$, leading to an inverted $L$ and wrong conclusion.
  - Why it fails: Students mix up the order when writing quickly, especially for power series problems.
  - Correct: Always explicitly label $a_n$ and $a_{n+1}$ first, then write $a_{n+1}$ in the numerator and $a_n$ in the denominator before simplifying.
- **Wrong:** Forgetting to take the absolute value of the ratio, leading to a negative $L$ and incorrect conclusion that $L < 1$ even for divergent series.
  - Why it fails: Students drop the absolute value for positive-term series and forget the habit for alternating/power series.
  - Correct: Always write the absolute value around the ratio before simplifying, even if all terms are positive, to build consistency.
- **Wrong:** Concluding convergence or divergence when $L=1$, without using a second test.
  - Why it fails: Students assume the ratio test can answer all questions and forget the inconclusive rule.
  - Correct: As soon as you get $L=1$, write "The ratio test is inconclusive" on your paper, then select another appropriate test.
- **Wrong:** Leaving $x$ inside the limit when calculating radius of convergence, leading to incorrect simplification.
  - Why it fails: Students forget $x$ is a constant with respect to the limit over $n$.
  - Correct: Always factor all terms involving $x$ out of the limit before evaluating.
- **Wrong:** Treating $L = \infty$ as an inconclusive case instead of a divergent case.
  - Why it fails: Students only remember $L<1$ and $L=1$ and forget that infinite $L$ falls into the $L>1$ category.
  - Correct: If the limit goes to infinity, note that $\infty > 1$, so the series diverges by the ratio test.

## Cheatsheet

| Category | Formula/Rule | Notes |
| --- | --- | --- |
| Core Ratio Test Limit | $L = \lim_{n \to \infty} \left\| \frac{a_{n+1}}{a_n} \right\|$ | For any infinite series $\sum a_n$; $a_{n+1}$ always in numerator |
| Convergence Case | $L < 1$ | Series converges absolutely, so it converges |
| Divergence Case | $L > 1$ (includes $L = \infty$) | Series diverges, $\lim_{n \to \infty} a_n \neq 0$ |
| Inconclusive Case | $L = 1$ | Use another test: p-test, limit comparison, alternating series test |
| Radius of Convergence | $R = \frac{1}{\lim_{n \to \infty} \left\| \frac{c_{n+1}}{c_n} \right\|}$ | Check endpoints $x = a \pm R$ separately |
| Infinite Radius | $R = \infty$ when $\lim_{n \to \infty} \left\| \frac{c_{n+1}}{c_n} \right\| = 0$ | Converges for all real $x$ |
| Zero Radius | $R = 0$ when $\lim_{n \to \infty} \left\| \frac{c_{n+1}}{c_n} \right\| = \infty$ | Converges only at center $x = a$ |
| Power Series Rule | Converges for $\|x - a\| < R$, diverges for $\|x - a\| > R$ | Interval of convergence only includes endpoints if they pass a separate test |

## What's next

Mastering the ratio test is a critical prerequisite for finding the full interval of convergence of power series, which is a 3-5 point question on almost every AP Calculus BC FRQ section. Without correctly applying the ratio test to find the radius of convergence, you cannot complete the interval of convergence, so this topic is make-or-break for a high Unit 10 score. The ratio test also prepares you for the root test, another convergence test, and underpins analysis of Taylor and Maclaurin series, the core of the second half of Unit 10.

- [Determining absolute or conditional convergence](https://www.owlsprep.com/study/ap-calculus-bc-u10-determining-absolute-or-conditional-convergence/)
- [Alternating Series Error Bound](https://www.owlsprep.com/study/ap-calculus-bc-u10-alternating-series-error-bound/)
- [Finding Taylor polynomial approximations of functions](https://www.owlsprep.com/study/ap-calculus-bc-u10-finding-taylor-polynomial-approximations-of/)

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