Condition

1. The ratio test is inconclusive $\lim_{n→\infin}|\frac {a_{n+1}} {a_n}|=1$

Let $\sum_{n=1}^\infin a_n$ be a series of nonzero terms.

Result

1. $\sum_{n=1}^\infin a_n$ converges if $\lim_{n→\infin}|\frac {a_{n+1}} {a_n}|<1$
2. $\sum_{n=1}^\infin a_n$ diverges if $\lim_{n→\infin}|\frac {a_{n+1}} {a_n}|>1$

• Reference and proof