10.8 Ratio Test for Convergence

The Ratio Test checks convergence by taking $L = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. If $L<1$, the series converges absolutely; if $L>1$, it diverges; and if $L=1$, the test tells you nothing and you need a different test. For AP Calculus BC, use this test first when the series has factorials or exponential terms.

How Does the Ratio Test Work?

The Ratio Test compares the size of consecutive terms. For a series $\sum a_n$, compute $L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$; the series converges absolutely if $L<1$, diverges if $L>1$, and needs another AP-approved test if $L=1$.

The Ratio Test is a BC-only topic, so AP Calculus AB students can skip it. On the BC exam it is one of the assessed convergence tests and the standard tool for finding the radius and interval of convergence of a power series, which comes later in this unit.

Comparing L to 1 and Setting Up the Ratio

• Use the formula $L = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ and compare $L$ to 1.
• $L < 1$ means the series converges absolutely; $L > 1$ means it diverges; $L = 1$ means the test is inconclusive.
• Reach for the Ratio Test first when a series has factorials ($n!$) or exponential terms ($a^n$).
• Build $a_{n+1}$ by replacing every $n$ with $(n+1)$, then divide and cancel carefully.
• Always take the absolute value before deciding, so a negative limit still gives a positive $L$.
• If you get $L = 1$, switch to another test from the unit (comparison, integral, alternating series, etc.).

What is the Ratio Test?

A ratio relates two numbers as a quotient. The Ratio Test looks at the ratio of one term to the term before it as $n$ grows large. If terms shrink fast enough, the series converges.

For a series $\sum a_n$, let

$$L = \lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$$

• If $L < 1$, the series converges (absolutely).
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

The general procedure is the same every time: identify $a_n$, write $a_{n+1}$ by replacing $n$ with $(n+1)$, set up the ratio, cancel common factors, and take the limit.

Working Ratio Test Problems Step by Step

Problem Solving: Work through these examples after trying them yourself.

Example 1: a series with a factorial

Determine whether $\sum \frac{5^n}{n!}$ converges or diverges.

Here $a_n = \frac{5^n}{n!}$. Find $a_{n+1}$ by replacing $n$ with $(n+1)$:

$$a_{n+1} = \frac{5^{n+1}}{(n+1)!} = \frac{5^n \cdot 5}{(n+1) \cdot n!}$$

Set up the ratio:

$$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n \to \infty} \frac{5^n \cdot 5}{(n+1) \cdot n!} \cdot \frac{n!}{5^n}$$

Cancel $5^n$ and $n!$:

$$\lim\limits_{n \to \infty} \frac{5}{n+1} = 0$$

Since $L=0<1$, the series converges absolutely by the Ratio Test.

Example 2: exponential terms

Determine whether $\sum \frac{4^{2n+1}(n+1)}{(-10)^n}$ converges or diverges.

Rewrite $a_n$:

$$a_n = \frac{4^{2n+1}(n+1)}{(-10)^n} = \frac{16^n \cdot 4 \cdot (n+1)}{(-10)^n}$$

Find $a_{n+1}$ by plugging $(n+1)$ into every $n$:

$$a_{n+1} = \frac{4^{2(n+1)+1}((n+1)+1)}{(-10)^{(n+1)}} = \frac{64 \cdot 16^n (n+2)}{(-10)^n \cdot (-10)}$$

Set up and simplify the ratio, canceling $(-10)^n$ and $16^n$:

$$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n \to \infty} \frac{16(n+2)}{-10(n+1)} = \lim\limits_{n \to \infty} \frac{16n+32}{-10n-10}$$

The limit of this ratio of leading terms is:

$$\lim\limits_{n \to \infty} -\frac{16}{10} = -\frac{8}{5}$$

The test uses the absolute value, so $L = \frac{8}{5}$. Since $\frac{8}{5} > 1$, the series diverges by the Ratio Test.

Example 3: the inconclusive case

Determine whether $\sum \frac{n+1}{n-2}$ converges or diverges.

Here $a_n = \frac{n+1}{n-2}$, and

$$a_{n+1} = \frac{(n+1)+1}{(n+1)-2} = \frac{n+2}{n-1}$$

Set up the ratio:

$$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n \to \infty} \left( \frac{n+2}{n-1} \cdot \frac{n-2}{n+1} \right) = 1$$

Since $L = 1$, the Ratio Test is inconclusive. You cannot tell from this test whether the series converges or diverges, so you would need another test. (A quick check: the terms here approach 1, not 0, so the nth term test actually shows this series diverges.)

Common Trap: When the limit comes out to a number with a sign, take the absolute value before comparing to 1. A limit of $-\frac{8}{5}$ becomes $L = \frac{8}{5}$, which is greater than 1, so the series diverges.

Common Misconceptions

• Misconception: L = 1 means the series converges.
  Actually: $L = 1$ does not mean the series converges. It means the test gives no information. Pick a different test instead.
• Misconception: You can omit the absolute value when applying the Ratio Test.
  Actually: A negative ratio can fool you. Always evaluate $\left| \frac{a_{n+1}}{a_n} \right|$ before deciding.
• Misconception: The Ratio Test is only for factorials.
  Actually: The Ratio Test is not only for factorials. It also handles exponential terms like $a^n$ and is the go-to tool for the radius of convergence of a power series. It tends to be weak for plain rational functions, where it often gives $L = 1$.
• Misconception: When building $a_{n+1}$, you only need to replace the obvious $n$ with $(n+1)$.
  Actually: Replace every $n$ with $(n+1)$, including inside exponents and factorials, not just the obvious spot.
• Misconception: "Converges" here means ordinary convergence.
  Actually: "Converges" here means absolute convergence. When $L < 1$, the series converges absolutely, which is stronger than ordinary convergence.

Closing

The Ratio Test rounds out your toolkit of convergence tests for this unit, which also includes the nth term test, integral test, p-series and harmonic series, comparison and limit comparison tests, and the alternating series test. With practice you will start to recognize which test fits a given series, and factorials or exponentials are usually a signal to try the Ratio Test first.

Related AP Calculus Guides

• Unit 10 Review: Infinite Series and Sequences
• 10.1 Defining Convergent and Divergent Infinite Series
• 10.3 The nth Term Test for Divergence
• 10.5 Harmonic Series and p-Series
• 10.2 Working with Geometric Series
• 10.4 Integral Test for Convergence
• 10.13 Radius and Interval of Convergence of Power Series