What does the root test say?

If lim sup (|a_n|)^{1/n} = L, then L 1 means divergence, L = 1 means inconclusive.

When should I use the root test instead of the ratio test?

Use the root test when the nth term contains an nth power, like (something)^n or n^n. The ratio test works better for factorials and exponentials.

What does 'inconclusive' mean?

It means the test does not decide convergence or divergence. You must use a different test, like the comparison test or integral test.

Can the root test be used on alternating series?

Yes. Take the absolute value of the terms before applying the root test. If the root test shows absolute convergence, the series converges absolutely.

How does the root test relate to power series?

The root test can find the radius of convergence of a power series. For Σ c_n (x-a)^n, R = 1 / lim sup |c_n|^{1/n}. Endpoints are tested separately.

Root Test for Series: Formula and Worked Examples

The Root Test

You have a series and you need to know if it converges. If the terms contain an nth power, the root test is your tool. The root test uses the nth root of the absolute value of the nth term: take lim sup (|a_n|)^{1/n} = L. If L < 1 the series converges absolutely; if L > 1 it diverges; if L = 1 the test is inconclusive.

When the Root Test Is the Natural Choice

Use the root test when a_n contains an expression raised to the nth power. The nth root cancels the exponent, leaving a limit you can evaluate. This is common with terms like (something)^n, n^{something}, or a combination of both.

If the root test gives L = 1, switch to another test. The ratio test is a frequent backup, but a comparison test or limit comparison test may be more useful when the root test fails.

Worked Examples

Example 1: Σ (n/(2n+1))^n

Apply the root test. Take (|a_n|)^{1/n} = n/(2n+1). The limit as n → ∞ is 1/2. Because 1/2 < 1, the series converges absolutely. Source: Stewart, Calculus, section 11.6, and Paul's Online Math Notes, Root Test.

Example 2: Σ (1 + 1/n)^{n^2}

Take the nth root: (|a_n|)^{1/n} = (1 + 1/n)^n. The limit is e > 1. The series diverges.

Example 3: Σ (2n+1)/(5n+3)^n

Take the nth root: ((2n+1)/(5n+3)^n)^{1/n} = (2n+1)^{1/n} / (5n+3). The numerator tends to 1, denominator tends to infinity. Limit is 0 < 1. The series converges.

Useful Limits

Two limits appear repeatedly when applying the root test:

n^{1/n} → 1 as n → ∞
(a_n)^{1/n} → 1 when a_n is a rational function of n (e.g., (n+1)/(2n+3))

These limits mean the root test is often inconclusive for rational-function terms. Use the ratio test or a comparison test for those.

Root Test vs Ratio Test

The ratio test uses |a_{n+1}/a_n|. The root test uses (|a_n|)^{1/n}. They give the same result when both work, but one may be easier to compute. The root test is decisive for terms with nth powers; the ratio test is better for factorials and exponentials. Both are inconclusive when L = 1.

For the series Σ n!/n^n, the ratio test is messy; the root test gives L = 1/e < 1, so the series converges. For Σ (2n+1)/(3n+2), both tests give L = 1 and are inconclusive; use the limit comparison test instead.

The root test is from Stewart, Calculus, section 11.6, and Paul's Online Math Notes, Root Test. The ratio test is from the same sources.

Root Test Calculator

Enter the nth term of the series. The calculator computes the limit of (|a_n|)^{1/n} and reports L. If L < 1, the series converges; if L > 1, it diverges; if L = 1, the test is inconclusive. Use it to verify manual calculations.

Nth Root Test

The nth root test is another name for the root test. The 'nth root' refers to taking the nth root of the absolute value of the nth term. The procedure is the same: compute lim sup (|a_n|)^{1/n} and compare L to 1.