Why do some series converge and some series diverge?

Question

What is the difference between convergent and divergent series? How is a convergent series sum related to the limit of the series? How are asymptotes related to limits? As series become infinite, how do the sums of convergent series differ from the sums of divergent series? How can we ever know what happens to a function or series as it approaches infinity?

Answer 1

Wikipedia is your friend:
http://en.wikipedia.org/wiki/Convergent_series