Root Test - Expii

By comparing to an infinite geometric series, we get a convergence test called the root test. Namely, for the series \(\sum a_n\), consider the sequence of \(n\)th roots: \(|a_n|^{\frac{1}{n}} = \sqrt[n]{|a_n|}\). If this "root sequence" converges to a number \(L<1\), then the series \(\sum a_n\) converges absolutely. If the root sequence converges to a number \(L>1\), then the series \(\sum a_n\) diverges. Otherwise, if the limit \(\lim_{n\to\infty} |a_n|^{\frac{1}{n}}\) either equals \(1\) or does not exist, then the test is inconclusive.