# 9.6 | The Ratio and Root Tests

Ratio Test
Let $$\sum a_n$$ be a series with nonzero terms.

1. The series $$\sum a_n$$ converges absolutely when $$\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1$$.
2. The series $$\sum a_n$$ diverges when $$\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1$$.
3. The Ratio Test is inconclusive when $$\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1$$.
Root Test

1. The series $$\sum a_n$$ converges absolutely when $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} <1$$.
2. The series $$\sum a_n$$ diverges when $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} >1$$.
3. The Root Test is inconclusive when $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} = 1$$.