MTH 251 — Calculus II
11.6 Absolute Convergence and the Ratio and Root Test
March 20, 2026
1. Learning Objectives
Upon completion of this section, you will be able to
- use the ratio test to check the convergence of series, and
- use the root test to check the convergence of series.
2. The Ratio Test
Theorem. Let \(\sum a_n\) be an infinite series and suppose
\begin{align*} L&=\lim _{n\to \infty }\left |\frac {a_{n+1}}{a_n}\right | \end{align*}
Then,
-
a)
- If \(L<1\), then \(\sum a_n\) is absolutely convergent.
-
b)
- If \(L>1\), then \(\sum a_n\) is divergent.
-
c)
- If \(L=1\), then the test is inconclusive (i.e. you need to use other tests).
Remark. Recall \(k!=1\cdot 2\cdot \cdots \cdot (k-1)\cdot k\). We often use \((k+1)!=k!(k+1)\).
3. The Root Test
Theorem (Root Test). Let \(\sum a_n\) be a series and suppose
\begin{align*} L&=\lim _{n\to \infty }\sqrt [n]{|a_n|} \end{align*}
Then,
-
a)
- If \(L<1\), then \(\sum a_n\) is absolutely convergent.
-
b)
- If \(L>1\), then \(\sum a_n\) is divergent.
-
c)
- If \(L=1\), then the test is inconclusive.