Calculus II: Ratio and Root Tests

In engineering and advanced sciences, you often encounter infinite series representing everything from signal approximations to error terms in numerical methods. Determining whether these series converge (sum to a finite value) or diverge (grow without bound) is not just academic—it tells you if a model is stable or a calculation is valid. While tests like the integral or comparison tests are useful, the Ratio Test and Root Test are particularly powerful for series whose terms involve factorials, exponentials, or nth powers, making them indispensable tools for your toolkit.

The Ratio Test: Analyzing Growth Rates

The Ratio Test is your go-to for series where each term involves multiplication or division by a growing factor, like factorials or exponential functions. It analyzes the limit of the ratio of successive terms.

Formally, for a series $\sum a_n$, you consider the limit: $$L=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|.$$ The test gives three possible 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. The series may converge or diverge, and you must use another test.

Let's apply it to a classic series involving a factorial. Consider ${\sum }_{n=1}^{\infty }\frac{n!}{10^n}$.

1. Identify $a_n=\frac{n!}{10^n}$ and $a_{n+1}=\frac{(n+1)!}{10^{n+1}}$.
2. Form the absolute ratio:

$$\left|\frac{a_{n+1}}{a_n}\right|=\frac{(n+1)!}{10^{n+1}}\cdot \frac{10^n}{n!}=\frac{(n+1)\cdot n!}{10\cdot 10^n}\cdot \frac{10^n}{n!}=\frac{n+1}{10}.$$

1. Take the limit:

$$L=\underset{n\to \infty }{\lim }\frac{n+1}{10}=\infty .$$ Since $L>1$, the series diverges. The factorial grows explosively faster than the exponential $10^n$.

For a series with exponentials, like ${\sum }_{n=1}^{\infty }\frac{5^n}{n^2}$, you get: $$L=\underset{n\to \infty }{\lim }\left|\frac{5^{n+1}}{(n+1{)}^2}\cdot \frac{n^2}{5^n}\right|=\underset{n\to \infty }{\lim }5\cdot {\left(\frac{n}{n+1}\right)}^2=5.$$ Since $L=5>1$, this series also diverges. The Ratio Test excels here because the exponential factor $5^n$ simplifies cleanly in the ratio.

The Root Test: Examining nth Term Behavior

The Root Test is particularly effective when the nth term of the series is raised to the nth power, or more generally, involves an expression of the form $(\text{something}{)}^n$. It looks at the limit of the nth root of the absolute value of the term.

For the series $\sum a_n$, you calculate: $$L=\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}.$$ The convergence conclusions are identical to the Ratio Test:

1. $L<1$: Converges absolutely.
2. $L>1$ (or infinity): Diverges.
3. $L=1$: Inconclusive.

Consider the series ${\sum }_{n=1}^{\infty }{\left(\frac{3n+2}{5n+7}\right)}^n$. Here, the entire expression is raised to the nth power, signaling the Root Test.

1. Identify $a_n={\left(\frac{3n+2}{5n+7}\right)}^n$.
2. Take the nth root:

$$\sqrt[n]{|a_n|}=\sqrt[n]{{\left(\frac{3n+2}{5n+7}\right)}^n}=\frac{3n+2}{5n+7}.$$

1. Take the limit:

$$L=\underset{n\to \infty }{\lim }\frac{3n+2}{5n+7}=\frac{3}{5}.$$ Since $L=3/5<1$, the series converges absolutely.

The power of the Root Test is that it directly "cancels" the outer nth power. Try it on ${\sum }_{n=1}^{\infty }\frac{2^n}{n^n}$. You get: $$L=\underset{n\to \infty }{\lim }\sqrt[n]{\frac{2^n}{n^n}}=\underset{n\to \infty }{\lim }\frac{2}{n}=0.$$ Since $L=0<1$, it converges. The Ratio Test would also work here, but the algebra is more cumbersome.

When the Tests Fail and How to Compare Them

A critical skill is recognizing when these tests are inconclusive, which happens precisely when the limit $L=1$. This occurs frequently with rational functions of $n$. For example, for the p-series $\sum \frac{1}{n^2}$ (which converges) and the harmonic series $\sum \frac{1}{n}$ (which diverges), both the Ratio and Root Tests yield $L=1$. When you get $L=1$, you must pivot to other tests like the Integral Test, Comparison Test, or Limit Comparison Test.

While both tests share the same three possible outcomes, their effectiveness differs depending on the series structure.

• The Ratio Test is often simpler for series containing factorials or products of terms like $n!$, $(2n)!$, or $3\cdot 7\cdot 11\cdot ...\cdot (4n-1)$. The successive terms naturally divide, simplifying the algebra.
• The Root Test is often superior when the nth term involves an expression raised to the nth power, like $(b_n{)}^n$, as we saw. It is also the only practical choice when the term involves multiple exponents, like $n^n$.

For series with both exponentials and powers, like $\sum \frac{x^n}{n^k}$, the Ratio Test is typically cleaner. You'll find the Ratio Test is more commonly used in practice for power series (used in engineering to solve differential equations) because of how neatly it handles the $x^n$ term.

Common Pitfalls

Even with straightforward rules, several mistakes can trip you up.

1. Misapplying the Test to the Wrong Series: Attempting the Ratio Test on a series like $\sum (\frac{1}{n}{)}^n$ is inefficient when the Root Test gives an instant answer. Conversely, using the Root Test on a series with factorials leads to a messy limit involving factorials and nth roots. Analyze the term's structure first to choose the most efficient tool.

1. Forgetting Absolute Values: The tests require the absolute value of the ratio or the nth root. For series with all positive terms, this is irrelevant. But for series with alternating signs or more complex sign patterns, omitting the absolute value bars means you are not applying the test correctly and may get an incorrect result regarding absolute convergence.

1. Incorrect Limit Calculation: The most common algebraic error in the Ratio Test is incorrectly forming $a_{n+1}$. For $a_n=\frac{n^2}{2^n}$, remember that $a_{n+1}=\frac{(n+1{)}^2}{2^{n+1}}$, not $\frac{n^2+1}{2^n+1}$. For the Root Test with $a_n={\left(1+\frac{3}{n}\right)}^{n^2}$, you must handle the exponent carefully: $\sqrt[n]{a_n}={\left(1+\frac{3}{n}\right)}^n$, whose limit is $e^3$, not $1$.

1. Misinterpreting $L=1$: A result of $L=1$ is not "maybe convergent." It is a definitive statement that the test fails and gives no information. You must immediately stop using that test and switch strategies. Do not try to argue that because $L$ is "close to" 1, the series probably converges—this is mathematically invalid.

Summary

• The Ratio Test analyzes ${\lim }_{n\to \infty }|a_{n+1}/a_n|$ and is highly effective for series with factorials, exponentials, or products where successive terms simplify upon division.
• The Root Test analyzes ${\lim }_{n\to \infty }\sqrt[n]{|a_n|}$ and is the preferred method for series whose general term is raised to the nth power, $(b_n{)}^n$.
• Both tests conclude absolute convergence for $L<1$, divergence for $L>1$, and are inconclusive for $L=1$, requiring a fallback to tests like Comparison or Integral.
• Choosing between them depends on the series structure: use the Ratio Test for multiplicative terms and the Root Test for exponential/nth-power terms. Mastering both allows you to efficiently tackle a wide array of series central to engineering analysis and applied mathematics.