12. Which of the following are sufficient conditions for the convergence of

A. None

B. I only

C. II only

D. III only

E. The correct answer is not given by A, B, C, or D.

 

Solution: The answer is D

 

Condition I: Is not a sufficient condition for convergence.

Reason: Consider the following series,

,

The nth term goes to 0:

But the series diverges

Condition II: Is not a sufficient condition for convergence. Consider then series

The Nth partial sum SN , is

Condition III: Is a sufficient condition for convergence for an absolutely convergent series converges.