The root test is a powerful tool in calculus for deciding whether an infinite series converges or diverges by examining the behavior of the nth‑root of its terms. When you apply the root test, you compute the limit superior of the nth‑root of the absolute value of the general term; if this limit is less than one the series converges absolutely, if it is greater than one the series diverges, and if it equals one the test is inconclusive. In real terms, this method is especially useful for series whose terms involve powers, factorials, or exponential expressions, because taking the nth‑root often simplifies the expression dramatically. In the following sections we will walk through the theory behind the root test, outline a step‑by‑step procedure for its application, illustrate the method with several examples, and answer common questions that arise when students first encounter this convergence criterion.
How the Root Test Works
Formal Statement
Consider an infinite series (\displaystyle \sum_{n=1}^{\infty} a_n). Define
[ L = \limsup_{n\to\infty} \sqrt[n]{|a_n|}. ]
Then:
- If (L < 1), the series converges absolutely.
- If (L > 1) (or (L = \infty)), the series diverges.
- If (L = 1), the root test gives no information; other tests must be used.
The use of (\limsup) (limit superior) ensures the test works even when the ordinary limit does not exist, as long as the upper bound of the nth‑roots settles to a particular value Simple, but easy to overlook. Still holds up..
Why Taking the nth‑Root Helps
For many series, the term (a_n) contains an exponent that grows with (n), such as (r^n), (n!Taking the nth‑root essentially “undoes” that growth, converting exponential behavior into a constant or a simpler expression. Here's the thing — ), or ((\ln n)^n). Here's one way to look at it: (\sqrt[n]{r^n}=|r|) and (\sqrt[n]{n!}) behaves roughly like (\frac{n}{e}) (by Stirling’s approximation), making the limit easier to evaluate.
Step‑by‑Step Procedure
Follow these steps whenever you want to apply the root test to a series (\sum a_n):
- Identify the general term (a_n). Write it in a form that highlights powers, factorials, or exponentials.
- Form the absolute value (|a_n|). The root test concerns absolute convergence, so drop any alternating signs.
- Take the nth‑root: compute (\sqrt[n]{|a_n|}). Simplify the expression as much as possible (use properties of radicals, logarithms, or known limits).
- Evaluate the limit (L = \lim_{n\to\infty} \sqrt[n]{|a_n|}). If the ordinary limit does not exist, compute the limit superior instead.
- Apply the decision rule:
- (L < 1) → convergence (absolute).
- (L > 1) → divergence.
- (L = 1) → test inconclusive; try another test (ratio, comparison, integral, etc.).
- State your conclusion clearly, referencing the value of (L) you found.
Worked Examples
Example 1: Geometric‑type Series
Determine the convergence of (\displaystyle \sum_{n=1}^{\infty} \left(\frac{3}{4}\right)^n).
- General term: (a_n = \left(\frac{3}{4}\right)^n).
- Absolute value: (|a_n| = \left(\frac{3}{4}\right)^n) (already positive).
- nth‑root: (\sqrt[n]{|a_n|} = \sqrt[n]{\left(\frac{3}{4}\right)^n} = \frac{3}{4}).
- Limit: (L = \lim_{n\to\infty} \frac{3}{4} = \frac{3}{4}).
- Since (L = 0.75 < 1), the series converges absolutely.
(Notice that this series is actually geometric; the root test reproduces the familiar condition (|r|<1).)
Example 2: Factorial‑heavy Series
Test (\displaystyle \sum_{n=1}^{\infty} \frac{n!}{n^n}).
- General term: (a_n = \frac{n!}{n^n}).
- Absolute value: same, as terms are positive.
- nth‑root:
[ \sqrt[n]{|a_n|} = \sqrt[n]{\frac{n!}{n^n}} = \frac{\sqrt[n]{n!}}{n}. ]
Using Stirling’s approximation (n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n),
[ \sqrt[n]{n!} \approx \left(\sqrt{2\pi n}\right)^{1/n} \cdot \frac{n}{e} \approx \frac{n}{e}, ]
because (\left(\sqrt{2\pi n}\right)^{1/n} \to 1). Hence
[ \sqrt[n]{|a_n|} \approx \frac{n/e}{n} = \frac{1}{e}. ]
- Limit: (L = \lim_{n\to\infty} \frac{1}{e} = \frac{1}{e} \approx 0.3679).
- Since (L < 1), the series converges absolutely.
Example 3: Root Test Inconclusive
Examine (\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}) Small thing, real impact..
- General term: (a_n = \frac{1}{n}).
- Absolute value: same.
- nth‑root: (\sqrt[n]{|a_n|} = \sqrt[n]{\frac{1}{n}} = \frac{1}{\sqrt[n]{n}}).
- Limit: (\displaystyle \lim_{n\to\infty} \sqrt[n]{n} = 1), so (L = \frac{1}{1} = 1).
- Because (L = 1), the root test is inconclusive. (We know the harmonic series diverges by the integral test.)
Example 4: Divergent Series
Consider (\displaystyle \sum_{n=1}^{\infty} \left(\frac{n+1}{n}\right)^{n^2}).
- General term: (a_n = \left(\frac{n+1}{n}\right)^{n^2}).
- Absolute value: same (positive).
- nth‑root:
[ \sqrt[n]{|a_n|} = \left[\left(\frac{n+1}{n}\right)^{
As we analyze the sequence in question, the behavior of its nth root becomes crucial in determining convergence. But by evaluating the limit of the root, we find that the sequence either stabilizes near a particular value or grows uncontrollably. But in these cases, the decision hinges on whether the root approaches a fixed number less than, equal to, or greater than one. This decision directly guides us toward the ultimate fate of convergence.
Building on our previous insights, when the limit evaluates to a value strictly between 0 and 1, the series contracts; when it exceeds 1, it expands; and equality to 1 demands further scrutiny. Thus, applying the decision rule effectively streamlines our analysis across diverse scenarios.
To keep it short, the investigation reveals a clear path to convergence when the root test aligns with a value under unity, while divergence emerges elsewhere. Understanding these nuances strengthens our ability to predict infinite series outcomes.
Conclusion: Based on the limiting behavior derived, the sequence converges with a finite limit determined by the root test. The precise value underscores the importance of careful calculation in asymptotic analysis Worth keeping that in mind..
Conclusion: The limit evaluates to $\frac{3}{4}$, confirming a convergent series.
