Infinite series: Convergence tests
The ratio test
In some cases it is hard to calculate \( \alpha = \limsup_{n\to\infty} \sqrt[n]{|a_n|} \). For all positive sequences \(a_n \geq 0\) the following inequalities hold:
\[ \liminf_{n\to\infty}\ \frac{a_{n+1}}{a_n} \leq \liminf_{n\to\infty}\ \sqrt[n]{a_n} \leq \limsup_{n\to\infty}\ \sqrt[n]{a_n} \leq \limsup_{n\to\infty}\ \frac{a_{n+1}}{a_n} \]
When we apply this to the positive sequence \(|a_n|\) we can use the root test create a new test:
Ratio test Let \(\textstyle\sum a_n\) be a series.
(1) If \(\textstyle\limsup \left|\frac{a_{n+1}}{a_n}\right| < 1\) then the series converges.
(2) If \(\textstyle\liminf \left|\frac{a_{n+1}}{a_n}\right| > 1\) then the series diverges.
The ratio test is weaker then the root test: there are sequences where the root test does work but for which the ratio test doesn't give any information. Conversely, when the ratio test can be applied it is always possible to apply the root test instead. Nevertheless, this test may help when \(\textstyle\liminf \left|\frac{a_{n+1}}{a_n}\right|\) and \(\textstyle\limsup \left|\frac{a_{n+1}}{a_n}\right|\) are easier to calculate than \(\alpha\).
