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Simplifying Nth Roots for the Root Test

Simplify the nth-root expression carefully, then use its limit to determine what the root test can—and cannot—say about a series.

Advanced 8 min read

What You'll Learn

  • Rewrite nth roots of terms that contain an nth power.
  • Simplify absolute values before evaluating a root-test limit.
  • Use rational-function limits to find the root-test value.
  • Interpret root-test limits below and above one.
  • Recognize why a limit of one is inconclusive.
  • Identify common errors in simplifying nth roots.

Why Simplify the Nth Root First?

The root test examines the size of the \(n\)th root of the absolute value of each term in a series. In many problems, the term itself looks complicated, but taking its \(n\)th root removes a repeated power or makes a simpler expression visible.

The main task is therefore often not to calculate a difficult limit directly. It is to simplify the expression correctly first, then determine its limit. An incorrect simplification can change the value of the limit and lead to the wrong conclusion about the series.

Definition: For a series \(\sum_{n=1}^{\infty} a_n\), the root-test quantity at index \(n\) is \(r_n=\sqrt[n]{|a_n|}=|a_n|^{1/n}\). If \(r_n\) has a limit \(L\), the root test says: if \(L<1\), the series converges absolutely; if \(L>1\), the series diverges; and if \(L=1\), the test gives no conclusion.

The absolute value is essential: the test is about the magnitudes of the terms, including when the original terms have alternating or otherwise varying signs. The root test in this form requires that the limit of \(\sqrt[n]{|a_n|}\) exists. There are stronger versions using a limit superior, but they are not needed here.

The Basic Simplification

The most useful pattern occurs when a term is already an \(n\)th power. If \(u_n\) is real, then

$$ \sqrt[n]{|u_n^n|} = \bigl(|u_n|^n\bigr)^{1/n} = |u_n|. $$

This is an exact identity, not an approximation. The absolute value ensures that the quantity inside the root is nonnegative; the exponent laws then reduce the powers \(n\) and \(1/n\). If \(u_n\) is known to be nonnegative, the final absolute value can be omitted.

For example, if a term has the form \(a_n=(-1)^n(c_n)^n\), then \(|a_n|=|c_n|^n\), so its root-test quantity is \(|c_n|\). The sign factor disappears when we take the absolute value, and the repeated \(n\)th power disappears when we take the \(n\)th root.

The remaining limit may be straightforward. In particular, if the simplified expression is a quotient of two linear functions of \(n\), divide numerator and denominator by \(n\). This turns the limit into a quotient involving constants and terms that tend to zero.

Worked Example: A Limit Below One

Worked Example: Establishing Absolute Convergence

Consider

$$ \sum_{n=1}^{\infty}(-1)^{n+1} \left(\frac{2n+1}{4n+3}\right)^n. $$

Let \(a_n=(-1)^{n+1}\left(\frac{2n+1}{4n+3}\right)^n\). The fraction is positive for every \(n\geq 1\), and the sign factor has absolute value \(1\). Therefore

$$ |a_n| = \left(\frac{2n+1}{4n+3}\right)^n. $$

Now take the \(n\)th root. Since the base is positive, the power cancels exactly:

$$ \sqrt[n]{|a_n|} = \frac{2n+1}{4n+3}. $$

Divide the numerator and denominator by \(n\), then take the limit:

$$ \lim_{n\to\infty}\sqrt[n]{|a_n|} = \lim_{n\to\infty} \frac{2+\frac{1}{n}}{4+\frac{3}{n}} = \frac{2+0}{4+0} = \frac{1}{2}. $$

Because \(\frac{1}{2}<1\), the root test gives absolute convergence. As a check, the simplified root expression really does approach \(\frac{1}{2}\), and this value satisfies the test’s strict convergence condition.

Worked Example: A Limit Above One

Worked Example: Showing Divergence

Consider the series with positive terms

$$ \sum_{n=1}^{\infty} \left(\frac{3n+1}{2n+5}\right)^n. $$

Write \(a_n=\left(\frac{3n+1}{2n+5}\right)^n\). Its base is positive, so taking the \(n\)th root cancels the power:

$$ \sqrt[n]{|a_n|} = \frac{3n+1}{2n+5}. $$

Divide the numerator and denominator by \(n\):

$$ \lim_{n\to\infty}\sqrt[n]{|a_n|} = \lim_{n\to\infty} \frac{3+\frac{1}{n}}{2+\frac{5}{n}} = \frac{3}{2}. $$

Since \(\frac{3}{2}>1\), the root test says the series diverges. There is also a direct check on the terms: because the root-test quantity tends to \(\frac{3}{2}\), it is greater than \(\frac{5}{4}\) for all sufficiently large \(n\). Thus, for those indices, \(a_n>\left(\frac{5}{4}\right)^n\), so the terms do not tend to zero. This confirms divergence by the necessary condition that terms of a convergent series must tend to zero.

Worked Example: The Boundary Value One

Worked Example: Seeing What the Root Test Cannot Decide

First consider the convergent series \(\sum_{n=1}^{\infty}\frac{1}{n^2}\). Its root-test quantity is

$$ \sqrt[n]{\left|\frac{1}{n^2}\right|} = \left(\frac{1}{n^2}\right)^{1/n} = n^{-2/n}. $$

To evaluate the limit, take the natural logarithm of the positive expression \(n^{-2/n}\). Its logarithm is \(-2\frac{\ln n}{n}\), which tends to \(0\), since \(\frac{\ln n}{n}\to 0\). Therefore

$$ \lim_{n\to\infty}n^{-2/n} = e^0 = 1. $$

The root test is inconclusive. The series nevertheless converges, as the \(p\)-series \(\sum 1/n^p\) converges for \(p>1\).

Now consider the harmonic series \(\sum_{n=1}^{\infty}\frac{1}{n}\). Its root-test quantity is

$$ \sqrt[n]{\frac{1}{n}} = n^{-1/n}, \qquad \lim_{n\to\infty}n^{-1/n} = e^0 = 1. $$

Again the root test is inconclusive. This series diverges, as established by the harmonic-series result. The two examples have the same root-test limit but different convergence outcomes. Thus \(L=1\) cannot be interpreted as either convergence or divergence.

A Step-by-Step Method

Use the following process when the terms contain powers or other expressions whose roots can be simplified. The simplification comes before deciding which root-test case applies.

1
Identify the term.
Write the general term as \(a_n\), and note any signs or factors that change when you take \(|a_n|\).
2
Form the root-test quantity.
Calculate \(\sqrt[n]{|a_n|}\), keeping the absolute value in place until the magnitude is clear.
3
Simplify before taking a limit.
Use exponent laws to cancel an \(n\)th power with the \(n\)th root, and reduce any remaining expression that permits an easier limit.
4
Evaluate the limit.
For a quotient of polynomials of the same degree, divide the numerator and denominator by the highest power of \(n\).
5
Match the value to the test.
A limit below one gives absolute convergence; a limit above one gives divergence; a limit equal to one is inconclusive.

Why These Steps Work

The simplification step follows directly from the exponent laws. For any nonnegative \(x\), \(\sqrt[n]{x}=x^{1/n}\). When \(x=|u_n|^n\), the exponents multiply:

$$ \bigl(|u_n|^n\bigr)^{1/n} = |u_n|^{n(1/n)} = |u_n|. $$

This identity explains why an \(n\)th power is so useful in a root-test problem: it can turn the root-test quantity into the base itself. Once that has happened, ordinary limit methods can be used on the simpler sequence. The root test then compares its limiting value with \(1\), the boundary separating the two decisive cases.

The strict inequalities matter. A limit slightly below \(1\), such as \(\frac{1}{2}\), lies in the convergence case. A limit slightly above \(1\), such as \(\frac{3}{2}\), lies in the divergence case. Neither conclusion follows from the test when the limit is exactly \(1\).

Common Mistakes

One common error is to mishandle the exponent when taking the \(n\)th root. For example, the following calculation is wrong:

$$ \sqrt[n]{\left(\frac{2n+1}{4n+3}\right)^n} \ \mathrel{\not=}\ \left(\frac{2n+1}{4n+3}\right)^{1/n}. $$

The \(1/n\) power from the root multiplies the exponent \(n\), rather than replacing it:

$$ \left[ \left(\frac{2n+1}{4n+3}\right)^n \right]^{1/n} = \left(\frac{2n+1}{4n+3}\right)^{n(1/n)} = \frac{2n+1}{4n+3}. $$

Another mistake is to conclude that a series diverges whenever the root-test limit equals \(1\). The examples \(\sum 1/n^2\) and \(\sum 1/n\) show why that conclusion is invalid: both give the limit \(1\), but one converges and the other diverges. A value of \(1\) means only that this test does not decide the question; another argument is needed.

Why the Simplification Matters

The root test is particularly useful when the general term is naturally expressed as a quantity raised to the \(n\)th power. Before computing anything complicated, look for that structure and ask what happens to the absolute value and exponent after taking the root. In many such cases, the main limit becomes a familiar one, such as a quotient of linear expressions.

It is equally important to know when to stop. The root test provides a decisive conclusion only when its limit is strictly below or above one. At the boundary, a limit calculation may be completely correct while still leaving the convergence question unanswered. Recognizing that boundary is part of applying the test correctly.

Check Your Understanding

For each question, focus on simplifying the root-test quantity before interpreting its limit.

  1. For \(a_n=(-1)^n\left(\frac{n+2}{3n+1}\right)^n\), what is \(\sqrt[n]{|a_n|}\), and what limit does it approach?
  2. If \(\lim_{n\to\infty}\sqrt[n]{|a_n|}=\frac{4}{5}\), what conclusion does the root test give?
  3. If \(\lim_{n\to\infty}\sqrt[n]{|a_n|}=\frac{5}{4}\), what conclusion does the root test give?
  4. Why does a root-test limit of \(1\) not establish either convergence or divergence?
  5. Identify and correct the exponent error in claiming that \(\sqrt[n]{(3/7)^n}=(3/7)^{1/n}\).

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