Let`s define a sequence,

A sequence is an ordered list of numbers, where each number in the sequence is called term. When we talk about the limit of a sequence we can say that if the terms of a sequence approaches to a number; in the other hand if the terms of that sequence do not approach to any number, has no limit and in  this case the sequences diverges.

Here is an example of convergent sequence.

Irma is having a several pain her arm due to and accident, in order to decrease that pain she takes an amount of ketorolac, this medicament is washed out her bloodstream every hour the same percentage. Now,we know that every hour the amount of the drug will be reducing, this information give us the clue that the amount of ketorolac will be gradually lowered to near 0, when the drug is out of the body, here we can evidence an example in the real life of a sequence that has a logical order and satisfies the requirements of a  converget sequence.

Another example is the moon phases because we can see that depending on the days the moon will be full, the half or less  this changes have a pattern and go in order and are approaching to something, in this case to have a full moon or a new moon.

http://mrvillascienceclass.blogspot.ca/2016/01/moon-phases.html

Here is an example of divergent sequence.

 Carmen, has a cronical disease called Hyperthyroidism , she has to take a pill all days to control this disease, in a determined moment she is not feeling well with this pills, her doctor said that she has to take a pill with more concentration of the medicament to counter the effects. There will be a time when  Carmen needs a stronger medicament; we can evidente here that if the medicament increase, it will do in a logical way (has a pattern)   as well as quantity.  The amount of the medicament will be increasing gradually and has patron. In this case we show that this is divergent sequence.