## Summary of Ratio and Root Tests

### Essential Concepts

• For the ratio test, we consider

$\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|$.

If $\rho <1$, the series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges absolutely. If $\rho >1$, the series diverges. If $\rho =1$, the test does not provide any information. This test is useful for series whose terms involve factorials.

• For the root test, we consider

$\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}$.

If $\rho <1$, the series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ converges absolutely. If $\rho >1$, the series diverges. If $\rho =1$, 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 $p-\text{series,}$ consider one of the comparison tests.

## Glossary

ratio test
for a series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$ with nonzero terms, let $\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|$; if $0\le \rho <1$, the series converges absolutely; if $\rho >1$, the series diverges; if $\rho =1$, the test is inconclusive
root test
for a series $\displaystyle\sum _{n=1}^{\infty }{a}_{n}$, let $\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}$; if $0\le \rho <1$, the series converges absolutely; if $\rho >1$, the series diverges; if $\rho =1$, the test is inconclusive