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Root Test

Root Test

A convergence test used when series terms contain nth powers.

 

Root Test: Compute r = lim(n→∞) of nth root of |a_n|. 1. r<1: converges absolutely. 2. r>1: diverges. 3. r=1: inconclusive.

 

 

See also

Ratio test, convergent series, divergent series, absolute convergence, limit, nth root, infinity

Key Formula

r=lim⁡n→∞∣an∣n=lim⁡n→∞∣an∣1/nr = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} |a_n|^{1/n}
Where:
  • ana_n = The nth term of the series
  • rr = The limit value used to determine convergence
  • nn = The index of the term, approaching infinity

Worked Example

Problem: Determine whether the series converges or diverges: ∑n=1∞(3n4n+1)n\sum_{n=1}^{\infty} \left(\frac{3n}{4n+1}\right)^n
Step 1: Identify the nth term of the series.
an=(3n4n+1)na_n = \left(\frac{3n}{4n+1}\right)^n
Step 2: Take the nth root of the absolute value of the nth term. Since the entire expression is raised to the nth power, the nth root cancels the exponent.
∣an∣n=3n4n+1\sqrt[n]{|a_n|} = \frac{3n}{4n+1}
Step 3: Evaluate the limit as n approaches infinity. Divide numerator and denominator by n.
r=lim⁡n→∞3n4n+1=lim⁡n→∞34+1n=34r = \lim_{n \to \infty} \frac{3n}{4n+1} = \lim_{n \to \infty} \frac{3}{4 + \frac{1}{n}} = \frac{3}{4}
Step 4: Apply the Root Test conclusion. Since r = 3/4 < 1, the series converges.
r=34<1  ⟹  convergesr = \frac{3}{4} < 1 \implies \text{converges}
Answer: The series converges by the Root Test since r = 3/4 < 1.

Another Example

This example involves nth powers in both the numerator and denominator (not just one compound expression raised to the nth power), showing how the Root Test handles terms where the ratio of exponential growth rates determines convergence.

Problem: Determine whether the series converges or diverges: ∑n=1∞2nnn\sum_{n=1}^{\infty} \frac{2^n}{n^n}
Step 1: Identify the nth term.
an=2nnna_n = \frac{2^n}{n^n}
Step 2: Take the nth root of the absolute value of the nth term. Apply the nth root to both numerator and denominator separately.
∣an∣n=2nnnnn=2n\sqrt[n]{|a_n|} = \frac{\sqrt[n]{2^n}}{\sqrt[n]{n^n}} = \frac{2}{n}
Step 3: Evaluate the limit as n approaches infinity.
r=lim⁡n→∞2n=0r = \lim_{n \to \infty} \frac{2}{n} = 0
Step 4: Since r = 0 < 1, the Root Test tells us the series converges.
r=0<1  ⟹  convergesr = 0 < 1 \implies \text{converges}
Answer: The series converges by the Root Test since r = 0 < 1.

Frequently Asked Questions

What is the difference between the Root Test and the Ratio Test?
Both tests compute a limit r and use the same thresholds (r < 1 converges, r > 1 diverges, r = 1 inconclusive). The Root Test computes the nth root of |a_n|, while the Ratio Test computes the ratio |a_{n+1}/a_n|. The Root Test is generally preferred when terms contain nth powers, whereas the Ratio Test works better with factorials or products. Mathematically, whenever the Ratio Test gives a conclusive answer, the Root Test also gives the same answer, but not vice versa—so the Root Test is technically at least as powerful.
When is the Root Test inconclusive?
The Root Test is inconclusive when r = 1. This happens, for example, with p-series like 1/n^p, where the nth root of 1/n^p approaches 1 for any fixed p. In such cases, you must use a different test, such as the p-series test, comparison test, or integral test, to determine convergence.
Does the Root Test prove absolute convergence?
Yes. Because the Root Test uses |a_n| (the absolute value of each term), a conclusion of convergence from the Root Test actually establishes absolute convergence. This is a stronger result than conditional convergence and means the series converges regardless of the signs of its terms.

Root Test vs. Ratio Test

Root TestRatio Test
Formular = lim |a_n|^{1/n}r = lim |a_{n+1} / a_n|
Best used whenTerms involve nth powers, like (f(n))^nTerms involve factorials or recursive products
Convergence criterionr < 1: converges; r > 1: diverges; r = 1: inconclusiver < 1: converges; r > 1: diverges; r = 1: inconclusive
StrengthAt least as powerful as the Ratio Test (sometimes more so)May fail (give r = 1) in cases where Root Test succeeds
Proves absolute convergence?YesYes

Why It Matters

The Root Test appears frequently in calculus II and introductory analysis courses when you study infinite series. It is particularly important for determining the radius of convergence of power series, since the formula for the radius involves the same nth-root limit (this connection is formalized in the Cauchy–Hadamard theorem). Mastering the Root Test gives you a reliable tool for series whose terms have the structure of something raised to the nth power—a pattern that other tests handle less efficiently.

Common Mistakes

Mistake: Forgetting to take the absolute value of a_n before computing the nth root.
Correction: The Root Test requires |a_n|^{1/n}. Without the absolute value, you may get a negative number under an even root or reach an incorrect limit. Always include the absolute value.
Mistake: Concluding convergence or divergence when r = 1.
Correction: When r = 1, the Root Test is inconclusive—it tells you nothing. You must switch to another test. A common trap is the harmonic series (1/n), where the nth root of 1/n → 1, yet the series diverges.

Related Terms