What distinguishes convergent sequences from divergent sequences?

First, convergent sequences have a finite limit, and divergent sequences don not have one.

Which means that, when a sequence for example: 1.1, 1.01, 1.001, 1.0001,…. has a limit of 1, so the sequence converges to 1. On the other, a divergent sequence for example, 1,2,3,4,5,,6…. , the limit is infinity, which it is not a real number, so it does not converge.

What distinguishes convergent series from divergent series?

Different with sequence, series is the sum of the all the term of the sequence. If we add all the term to calculate the sum( the series) we got a number is too big, also means it don not have a finite value. Same idea, when the number in the series add up, if it have a finite number, it converges, it do not have( or close to a infinite ) it diverges.

Thoughts:

In my opinions, both convergent and divergent represent a idea or a way of thinking questions. When we think something is converge, we can find  only one solution to this question, and no other answer will appear. Just like, calculate the right number of the distance between Mars and Earth. However, when we dealing with a divergent questions, we can find many answer(infinite), like, how many stars in the this galaxy.