• what distinguishes convergent sequences from divergent sequences, and

1. The definition of consequence: A sequence {an}={a1,a2,a3…an} is ordered list of elements with a very element but no lost element .We say that limx→∞ an = L,or {an}converges to L. If an is arbitrarily close to L provided n is sufficiently large. A sequence that does not converge is said to diverge.
2. So we can say if a sequence is close to a number, it is convergent, but if a sequence is infinite, it is divergent.
3. When we look at a sequence, we can see sequence will have different kinds of highest or lowest points, like 0, 1, 2, 3, 4…….

If a sequence without a highest or lowest point even does not get close to a top or low point, it must be divergent.

When we see a graph of convergent sequence, take it as an example of an=sin(x)/x, a following graph show, it has many top and low point, and the points are continually close to a point which is the L as we said before. Also , if we consider the top and low points as that there are two friends who want to meet each other from the top and feet of mountain, and they decide to meet each other at a point between their positions by taking cable car. So the lower points will increase and the high points will decrease because of their meeting later. Of course, their distance become closer and closer.

On the otherwise, if a sequence is divergent, it is like that two people take cable car and their distance becomes larger and larger.(distance of cable car)

(b) what distinguishes convergent series from divergent series. Series are something sum of an, like =a1, a1+a2,a1+a2+a3… . If they are convergent, they can get a certain number as L, if they are divergent, they may be infinite. We have many types of test to test, like ratio test, comparison test, divergent test…..to decide which is divergent or converge. Also, we can use a simple word to understand. If you draw a graph, like {1/n^2), if you put cake there, you will find the size of the cake becomes smaller and smaller, but is you series like ={2^n}, you will find the size of cake becomes larger and larger.