The Root Test: Let $\rho = \displaystyle{\lim_{n \to \infty} |a_n|^{1/n}}$.

• If $\rho<1$, then $\sum a_n$ converges absolutely $\big($like the geometric series $\sum \rho^n$$\big)$


• If $\rho>1$, then $\sum a_n$ diverges $\big($like the geometric series $\sum \rho^n$$\big)$


• If $\rho=1$, or $\rho$ does not exist, then the test is inconclusive.