1. 9.6 Ratio and Root Tests

2. The Ratio Test
Ratio test works well for series that involve factorials and constants raised to nth power.

3. Examples

4. The Root Test: If is a series with positive terms and
then:
The series converges if
Note that the rules are the same as for the Ratio Test.
The series diverges if
The test is inconclusive if
Remark:
Root test works well for series that could be written as an nth power of the entire term.
If the ratio test fails (when L = 1), do not try Root Test because L will be 1 again.

5. Examples

6. Strategy to test a series
Use Divergence Test if
Recognize the special types of series: p-series, geometric series, telescoping series.
Use root test if the series involves nth power of the entire term.
Use ratio test if the series involves product of factorials, exponents, and polynomials.
Use limit comparison test if the series involves rational functions.
Use alternating series test if it is an alternating series and the above tests cannot be applied.
Use direct comparison test if the series has a similar form to a known series.
Use integral test if the general term can be integrated, and it’s the last test to try if all others fail.

7. Examples