Theorem 9.4.1. Ratio Test.
Let \(\{a_n\}\) be a positive sequence and consider \(\lim\limits_{n\to\infty}\dfrac{a_{n+1}}{a_n}\text{.}\)
- If \(\lim\limits_{n\to\infty}\dfrac{a_{n+1}}{a_n}\lt 1\text{,}\) then \(\ds\infser a_n\) converges.
- If \(\lim\limits_{n\to\infty}\dfrac{a_{n+1}}{a_n} \gt 1\) or \(\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=\infty\text{,}\) then \(\ds\infser a_n\) diverges.
- If \(\lim\limits_{n\to\infty}\dfrac{a_{n+1}}{a_n}=1\text{,}\) the Ratio Test is inconclusive.