# ratio test

Let $(a_{n})$ be a sequence of real numbers. Then the series

 $\sum_{n=1}^{\infty}a_{n}$
 $\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|<1$

and diverges if

 $\liminf_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|>1.$
Title ratio test RatioTest 2013-03-22 12:24:14 2013-03-22 12:24:14 Koro (127) Koro (127) 12 Koro (127) Theorem msc 26A06 msc 40A05 quotient criterion