When we use the ratio test, we find the limit:

\(\displaystyle L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\)

The ratio test states that:

- if \(L<1\) then the series converges absolutely;
- if \(L<1\) then the series is divergent;
- if \(L=1\) or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

And so, because we require \(L<1\), the interval of convergence is necessarily open, that is, we do not include the end-points.