(a) What distinguishes convergent sequences from divergent sequences
Convergent sequences: One day, a super virus appears. This super virus not only can make everything die, but also can eat everything to grow up. Because of the super virus unlimited devour, all the people, animals, plants even bacteria extinct. The super virus has nothing to eat. So, the only way for the super virus to keep alive is to eat itself. During the first day, half of the super virus eats the other half of the super virus. And in the second day, half of the half of the super virus eats the other half of the half of the super virus. The amount of the super virus in everyday is a convergent sequence.
Divergent sequence: From now on, let us calculate the time just pass by which is 1s, 2s, 3s…. Although we can continue calculate, there will have no end for the time. And the time is a divergent sequences.
(b) What distinguishes convergent series from divergent series
Convergent series: Image a young woman win 1,000,000,000$. She was so excited about this, but she was also too young to plan how to use these money. So she decided to use the half of the money for her whole life. But time is so fast. She runs out of the half of the money in one year. Because she also cannot believe the truth, she learns to plan the money. But she till runs out of the one fourth of the money in the second year. After many years, she still uses the half and the half and the half and …… of the 1,000,000,000$. The amount of the money that she used after she won the prize is a convergent series.
Divergent series: A person is so lazy. But he wants to build a stone house by himself. So he creates a plan: bring one stone back to home everyday. So, in the first day, this person gets one stone. And in the second day, this person gets one plus one which is two stones and so on. The amount of the stones that this person have is a divergent series.