But if p > 1 then converges, while if p 1 then diverges. This
illustrates the failure of the Ratio Test to resolve the question of convergence
when the limit of the ratio is 1.
THE ROOT TEST
Let , if it exists. Then converges if L < 1 and diverges if L > 1.
If L = 1 this test is inconclusive; try one of the other tests.
The decision rule for this test is the same as that for the Ratio Test.
NOTE: The Root Test is not specifically tested on the AP Calculus Exam;
however, we present it here because it may be helpful in determining
convergence.
Example 25 __
The series converges by the Root Test, since
B5. Alternating Series and Absolute Convergence
Any test that can be applied to a nonnegative series can be used for a series all of
whose terms are negative. We consider here only one type of series with mixed
signs, the so-called alternating series. This has the form:
where ak > 0. The series
is the alternating harmonic series.
BC ONLYTHE ALTERNATING SERIES TEST
An alternating series converges if:
(1) an (^) + 1 < an for all n, and (2) .