Essential Concepts
- For the ratio test, we consider
[latex]\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|[/latex].
If [latex]\rho <1[/latex], the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges absolutely. If [latex]\rho >1[/latex], the series diverges. If [latex]\rho =1[/latex], the test does not provide any information. This test is useful for series whose terms involve factorials. - For the root test, we consider
[latex]\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}[/latex].
If [latex]\rho <1[/latex], the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges absolutely. If [latex]\rho >1[/latex], the series diverges. If [latex]\rho =1[/latex], the test does not provide any information. The root test is useful for series whose terms involve powers. - For a series that is similar to a geometric series or [latex]p-\text{series,}[/latex] consider one of the comparison tests.
Glossary
- ratio test
- for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with nonzero terms, let [latex]\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|[/latex]; if [latex]0\le \rho <1[/latex], the series converges absolutely; if [latex]\rho >1[/latex], the series diverges; if [latex]\rho =1[/latex], the test is inconclusive
- root test
- for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex], let [latex]\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}[/latex]; if [latex]0\le \rho <1[/latex], the series converges absolutely; if [latex]\rho >1[/latex], the series diverges; if [latex]\rho =1[/latex], the test is inconclusive
