Consider a series \(\sum_{n=0}^{\infty} a_n\) with values in a normed space . Assume the limit \(L=\lim_{n \to \infty}\lvert \frac{a_{n+1}}{a_n}\rvert\) exists. If \(L<1\) then the series converges absolutely . Otherwise if \(L>1\) the series diverges.
This test is based on the convergence of the geometric series .