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.