Question 3:

For this question, we should use both the definition of limit and the definition of derivative. For part a and b, we need to use the definition of derivative to get two equations. For part c, we should find the relationship between the previous equations and try to get a new equation to solve this problem. Firstly, we use the definition of limit to write the definition of equations that are in part a and b. Then, we can find there exists a relationship between these two equations, so we can get a new definition from that. Then we can use the new definition to write a new equation, so we just get the answer.

1. (a). Convergent sequences have a finite limit. When n closes to infinity, An close to a finite number. eg. An=1/n. when n is close to infinity, limAn gets closer to 0.

(b). Divergent sequences do not have a finite limit. It means when n is close to infinity, An       close to +infinity or -infinity, and it does not close to a finite number.  eg. limAn=n. when n is close to infinity, An also gets closer to infinity.

2. (a). Convergent Series: When I add more and more terms of a convergent series, the sum of the series get closer to a certain number. It means the sum of the series have a finite limit. eg. {(2/3)^n} is a convergent series. 

(b). Divergent Series:  When I add more and more terms of a divergent series, the sum of the series cannot get closer to a certain number. Even if each partial sum gets closer to a certain number, the sum of the whole series get closer to +infinity or-infinity. eg. {1/n}is a divergent series, although limAn=1/n is a convergent sequence.