Other Convergence Tests

DEFINITION: An alternating series is a series whose terms alternate between positive and negative.



THEOREM (Alternating Series Test): An alternating series converges if the following two conditions are satisfied:

(a) 

(b) 

REMARK: Note that if an alternating series does not satisfy condition (b), then it diverges by the Divergence Test. On the other hand, if an alternating series satisfies (b) but does not satisfy (a), then it may converge or diverge. For example,



diverges, but



converges

EXAMPLES:

1. The series converges, since is a decreasing function and

2. The series converges, since is a decreasing function and

3. The series



diverges by the Divergence Test, since



4. Use the Alternating Series Test to determine whether the series



converge or diverge.


5. Use the Alternating Series Test to determine whether the series



converge or diverge.


DEFINITION: A series



is said to converge absolutely if the series of absolute values



converges and is said to converge conditionally if converges, but diverges.

EXAMPLES:

1. The series converges conditionally, since it converges, but diverges by the -test with

2. The series converges absolutely, since it converges and converges by the -test with

THEOREM: If the series



converges, then so does the series



THEOREM (The Ratio Test): Let be a series with nonzero terms and suppose that



(a) If then the series converges absolutely and therefore converges.

(b) If or if then the series diverges.

(c) If then the test is inconclusive which means the series may converge or diverge, so that another test must be tried.

EXAMPLE: Use the Ratio Test to determine whether the following series converge or diverge:

(a)


(b)


(c)


(d)


(e)


THEOREM (The Root Test): Let be any series and suppose that



(a) If then the series is absolutely convergent (and therefore convergent).

(b) If or then the series is divergent.

(c) If then the test is inconclusive which means the series may converge or diverge, so that another test must be tried.

EXAMPLE: Use the Root Test to determine whether the following series converge or diverge:

(a)


(b)


(c)


(d)