5.6 Ratio and Root Tests

Ratio and Root Tests for Series Convergence

The ratio and root tests help you determine whether an infinite series converges or diverges by measuring how fast the terms shrink. Both tests produce a single number, $L$, and the convergence decision follows the same rule for each: if $L < 1$ the series converges absolutely, if $L > 1$ it diverges, and if $L = 1$ the test tells you nothing.

These two tests are especially handy for series involving factorials, exponential terms, or terms raised to the $n$th power. When one test gives an inconclusive result, the other can sometimes break the tie.

The Ratio Test

Given a series $\sum_{n=1}^{\infty} a_n$, the ratio test examines how the size of each term compares to the previous one. You compute:

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

• If $L < 1$, the series converges absolutely. Each term is eventually shrinking by at least a fixed fraction, so the series behaves like a convergent geometric series.
• If $L > 1$ (or $L = \infty$), the series diverges. The terms aren't decreasing fast enough (or are actually growing).
• If $L = 1$, the test is inconclusive. You'll need a different test.

When to reach for the ratio test: Any time you see factorials ($n!$), products like $1 \cdot 3 \cdot 5 \cdots (2n-1)$, or exponential terms such as $a^n$. The ratio $\frac{a_{n+1}}{a_n}$ tends to simplify nicely in these cases because neighboring factorials and exponentials cancel.

Example. Test $\sum_{n=1}^{\infty} \frac{n!}{3^n}$ for convergence.

1. Write the ratio: $\left|\frac{a_{n+1}}{a_n}\right| = \frac{(n+1)!}{3^{n+1}} \cdot \frac{3^n}{n!}$
2. Simplify. The factorials cancel to leave $(n+1)$, and the powers of 3 cancel to leave $\frac{1}{3}$: $\frac{n+1}{3}$
3. Take the limit: $L = \lim_{n \to \infty} \frac{n+1}{3} = \infty$
4. Since $L > 1$, the series diverges.

The Root Test

For the same series $\sum_{n=1}^{\infty} a_n$, the root test instead looks at the $n$th root of each term's absolute value:

$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$

The decision rule is identical to the ratio test:

• $L < 1$: converges absolutely
• $L > 1$ (or $L = \infty$): diverges
• $L = 1$: inconclusive

When to reach for the root test: Series where the entire $n$th term is something raised to the $n$th power, like $\left(\frac{2n+1}{3n}\right)^n$. Taking the $n$th root peels off that outer exponent immediately, which is exactly what you want.

Example. Test $\sum_{n=1}^{\infty} \left(\frac{n}{2n+1}\right)^n$ for convergence.

1. Take the $n$th root: $\sqrt[n]{|a_n|} = \frac{n}{2n+1}$
2. Take the limit: $L = \lim_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2}$
3. Since $L = \frac{1}{2} < 1$, the series converges absolutely.

Trying the ratio test on this same series would require expanding $\left(\frac{n+1}{2n+3}\right)^{n+1} \Big/ \left(\frac{n}{2n+1}\right)^n$, which is far messier. That's a good sign you should use the root test instead.

Ratio vs. Root: Which One?

A quick rule of thumb:

• Factorials or products in the terms → try the ratio test first.
• Entire expression raised to the $n$th power → try the root test first.
• If one gives $L = 1$, try the other before moving on to a different test entirely.

Both tests detect absolute convergence, so if a series passes either test with $L < 1$, it converges absolutely (and therefore converges).

Systematic Approach to Choosing a Convergence Test

When you're staring at a series and aren't sure which test to use, work through this checklist:

1. Recognize known forms. Is it a geometric series ($|r| < 1$ converges) or a p-series ($p > 1$ converges)?
2. Divergence test. Compute $\lim_{n \to \infty} a_n$. If this limit isn't zero, the series diverges. (But a limit of zero does not guarantee convergence.)
3. Alternating series? If terms alternate in sign and decrease in absolute value toward zero, the alternating series test applies.
4. Ratio or root test. If the terms involve factorials, exponentials, or $n$th powers, apply the appropriate test as described above.
5. Comparison-based tests. If ratio/root are inconclusive ($L = 1$), compare the series to one you already know:
   • Direct comparison test: bound your series above or below by a known series.
   • Limit comparison test: compute $\lim_{n \to \infty} \frac{a_n}{b_n}$ where $\sum b_n$ is a known series. A finite, positive limit means both series share the same convergence behavior.
   • Integral test: if $a_n = f(n)$ for a positive, decreasing function $f$, the series and $\int_1^{\infty} f(x)\,dx$ converge or diverge together.

No single test handles every series. The goal is to build intuition for which test fits the structure of the terms you're looking at, and the ratio and root tests cover a large and common class of problems.