What does it mean for a series to converge or diverge?

[bmed90]

Currently, we are covering the topic of convergence and divergence of a series in my calculus 2 class. I was wondering if you could give me in there own words what it means for a series to converge, and what it means for a series to diverge.
I know that when a series converges, its limit reaches a definite value, and when it converges its limit is indefinite (infinity). However that right there is all I know of convergence and divergence. I am really trying to learn the concept behind convergence and divergence. If anyone would be willing to enlighten me with their own explanation of convergence and divergence it would be appreciated. Preferably in a more literal sense, rather than a mathematical explanation.

 

[HallsofIvy]

Saying that a series "converges" essentially means that, even though it contains an infinite number of terms, it has a "sum". (I am assuming that you really do mean "series" and not "sequence".) A series might diverge (not have a finite sum) because it is gets larger and larger ("goes to infinity"). But it is possible for a series to diverge even though it does NOT "go to infinity". For example, $\sum_{n=0}^\infty (-1)^n= 1- 1+ 1- 1+ \cdot\cdot\cdot$ diverges because its "partial sums", 1, 1- 1= 0, 1- 1+ 1= 1, 1- 1+ 1- 1= 0, ..., alternate between 0 and 1 and do not approach a single value.