Root test
= Root test
{title2=$r=\limsup |a_n|^{1/n}$}
{wiki}
For a series $\sum_n a_n$, let $r=\limsup_{n\to\infty}|a_n|^{1/n}$. If $r<1$, choose $r<\rho<1$; eventually $|a_n|\leq\rho^n$, so geometric comparison gives absolute convergence. If $r>1$, the terms fail to tend to zero along a subsequence and the series diverges. At $r=1$, the test is inconclusive.