The Root Test
The root test gives information about the convergence or not of a series by considering
Theorem (The Root Test).
If there exists some N such thatfor some q with n>=N thenconverges absolutely. If for infinitely manythendiverges.
Proof: Suppose there is a real numbersuch thatand a positive numbersuch that forthenand sinceconverges, hence by the comparison testconverges absolutely.
Suppose now thatfor infinitely manythen for infinitely manyso does not converge to zero anddoes not converge.
Example:even,odd. Even andodd. Thushence by the root test the series converges absolutely.
Example: Doesconverge for
soconverges to 1 soconverges toLetthenso there is a positive integersuch thatimplies thatthusconverges absolutely.
The root test has wider scope than the ratio test, since if the root test gives no results for convergence, then neither does the ratio test. Sometimes however the root test can be hard to use, and more information can usefully be obtained from the ratio test. Neither root nor ratio test will identify series that converge conditionally.