Expii

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.