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 \(\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} = 1\).

E 9.6 Exercises