Ratio Test - Expii

Sometimes the root test is hard to use, if \(\lim_{n\to\infty} |a_n|^{\frac{1}{n}}\) is hard to evaluate. However, the root test implies a weaker but simpler test called the ratio test. (In other words, the root test works whenever the ratio test works, but the ratio test might be easier to use.) Namely, if the ratio sequence \(|a_{n+1}/a_n|\) converges to a number \(r<1\), then the series \(\sum a_n\) converges absolutely. If the ratio sequence converges to a number \(r>1\), then the series \(\sum a_n\) diverges. Otherwise, if the limit \(\lim_{n\to\infty} |a_{n+1}/a_n|\) either equals \(1\) or does not exist, then the test is inconclusive.