Let $(a_n)$ be a sequence of real numbers. Then the series$$\sum_{n=1}^\infty a_n$$converges absolutely if $$\limsup_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|<1$$ and diverges if $$\liminf_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|>1.$$
40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
26A06 (Real functions :: Functions of one variable :: One-variable calculus)
