Ratio Test

Theorem:

Let ${\sum }_{n=1}^{\infty }{a}_n$ be a series with nonzero terms, and let

$\rho ={\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|$.

i. If $0\le \rho <1$, then the series converges absolutely.
ii. If $\rho >1$ or $\rho =\infty$, then the series diverges.
iii. If $\rho =1$, then the test does not provide any information.