In order to distinguish a convergent series from a divergent series, we have to know what makes a series converge.

First of all, a series is the sum of a sequence of numbers. Since I have already explained about convergence which is affected by boundary and monotone, we know that a convergent sequence will eventually get closer and closer to a certain number which we call L in the mathematical world. Similarly, with a series where the sum of the sequence is bounded and monotone and will get closer to L, it is defined as a convergent series. On the other hand, a divergent series is the sum of a sequence that does not get close to L.

An example of a convergent series is a person who is going on a one-mile walk. Let us assume that the person starts from zero position and zero time. The sequence is the distance that the person walks within a certain time. The series is the total distance that the person walks from the beginning. As we can see, the total distance is eventually getting closer and closer to one mile. Since this series is bounded between zero to one mile and is increasing and getting closer to one mile, it is convergent and converges to the L which equals to one in this case.

Similarly, an example of a divergent series is just like a person who does not know where to go. Since the person keeps going back and forth, the total distance is always changing and it never gets close to any point. This series diverges because it is not monotone and it does not reach to any L.

In conclusion, if a series or the sum of the sequence converges to L, then this series converges. Otherwise, it diverges.