Barrons AP Calculus - David Bock

EXAMPLE 23

Does converge or diverge?

SOLUTION:

and

(See §E2.) Since e > 1, diverges by the Ratio Test.

EXAMPLE 24

If the Ratio Test is applied to any p-series, then

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 nth 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.
Note that the decision rule for this test is the same as that for the Ratio Test.
EXAMPLE 25
The series converges by the nth 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.