A test similar in power to the Ratio Test, the Root Test is uniquely suited for series where the entire term is raised to the nth power. It analyzes the limit of the nth root of the terms.
State the formula used in the Root Test for a series \(\sum a_n\).
You must compute the limit \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} |a_n|^{1/n}\).
In the Root Test, if \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|} < 1\), what is the conclusion?
The series \(\sum a_n\) is absolutely convergent (and therefore converges).
In the Root Test, if \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|} > 1\) or \(L=\infty\), what is the conclusion?
The series \(\sum a_n\) diverges.
In the Root Test, if \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = 1\), what is the conclusion?
The test is inconclusive. The series might converge or diverge, and a different test must be used.
The Root Test is particularly effective for series with what kind of structure?
It is most effective for series where the general term \(a_n\) involves an expression raised to the \(n\)-th power, like \(a_n = (\dots)^n\).
Apply the Root Test to \(\sum_{n=1}^{\infty} \left( \frac{2n+3}{3n+2} \right)^n\).
Compute \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \left( \frac{2n+3}{3n+2} \right) = \frac{2}{3}\). Since \(L = 2/3 < 1\), the series converges.
What is the value of the useful limit \(\lim_{n \to \infty} \sqrt[n]{n}\)?
\(\lim_{n \to \infty} \sqrt[n]{n} = 1\). This is helpful when applying the Root Test to series like \(\sum \frac{1}{n^n}\).
Can the Root Test determine the convergence of the harmonic series \(\sum \frac{1}{n}\)?
No. \(L = \lim_{n \to \infty} \sqrt[n]{\frac{1}{n}} = \frac{1}{\lim_{n \to \infty} \sqrt[n]{n}} = \frac{1}{1} = 1\). Since the limit is \(1\), the Root Test is inconclusive.
