If we want to understand the distinguish between convergent series and divergent series, we first need to understand what is a series, in the former web page about the convergent sequences and divergent sequences, do you remember the example I give about the baby ducks. We get a sequence{1,2,3,4,….} . And a series is the sum of every figures in a sequence. For the sequence {1,2,3,4,….}, its series is sum=1+2+3+4+5+…+n .

And now let us understand what is convergent series and divergent series by reading a famous story. According to the legend,  a mathematician invented the chess and he was awarded by the emperor , When he was asked what reward he would want, After thinking, the mathematician wanted to put one rice in the first check of the chess, then put two rice in the second check, put four rice in the third check, put eight rice in the fourth check…. And so on, the number of the rice in every check is twice as the rice in former check. The emperor agreed the mathematician’s asking and wondered why the  mathematician ask for so little reward. But soon the emperor was in a big surprise. The country did not have enough rice to fulfill all checks. Because when the 64th check was fulfilled, there were 18446744073709551615 rice. And if it continues, it will be bigger and without a limit.

And in mathematics, we can transfer the figures in to a series{2+2^2+2^3+2^4+….+2^n} and it is also the sum of every figure in the sequence{2^n}. From the story ,we know it is infinity and without a limit. So we can define the series as a divergent series.

From the story above, we know use limit to identify a divergent series. And in contrast, when a series has a limit or a scale to bound it, we can define it as a convergent series.

If we want to understand the distinguish between convergent sequences and divergent sequences, we need to understand what is a sequence, in mathematics, sequence is an ordered list of numbers. I know it is a little abstract. So let us think it into our real life.

When we go to the lakeside,  sometimes we can see a harmonious scene, a mother duck lead some baby ducks swimming in the lake. And they keep a line one by one. How can we know how many baby ducks in the line. May be we always count 1,2,3, 4…until the last baby duck. The number we have counted can write down as {1,2,3,4…} and this is a sequence. And in mathematics, the figures in a sequence can be different.

Another we need t know is about the limit. Limit is the most important difference between convergent sequences and divergent sequences. Generally speaking, a convergent sequence has a limit, and a divergent sequence has no limit.

Let us return to the real life. What is limit… When you are having a long-distance running, there is a time that you feel you can not insist  the running, the distance you have insisted is your running limit. And in winter,  we are considered with the temperature and predict whether it will snow or not. That is because, when temperature drop to 0 celsius degree zero，the raindrop will become snow and therefore 0 celsius degree is a limit.

Let us connect examples with mathematics. There is a classic example, if we have a stick and assume it is enough long,first we separate it into two equal parts. And then separate one part of the stick into two new parts…And repeat the behavior…Finally, we can find that this behavior can continue till forever and it will never have an end. So we can say it has no limit. And in mathematics, we can use the sequence {(1/2)^n} to represent the left part of the stick. And we define it as a divergent sequence.

The separation of a stick is a classic example to understand the meaning of the divergent sequence. We  know the divergent sequence has no limit. And in contrast，if a sequence has a limit and so we can think it is a convergent sequence.

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