Summary of Ratio and Root Tests

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