A

16. Which of the following series converge?

A. I and II only	B. I and III only	C. II and III only
D. I, II, and III	E. The correct answer is not given by A, B, C, or D.

Solution: The answer is A

The above three series are alternating series. Thus, we can apply the Alternating Series Test to all three to determine which ones converge.

Series I: We have, for n > 1,

Thus a_n > a_n+1 and a_n -> 0 as n goes to infinity. The series converges.

Series II: We have

Thus a_n > a_n+1 and a_n -> 0 as n goes to infinity. The series converges

Series III: In this case

so the series does not converge.