divergent vs Convergent Series

[Guille]

What is the difference between divergent series and convergent series?

Is there any usefullness in these series, or are they pure mathematicall?

I've been analyzing both kind of series, and I understand them separatelly, more or less, but I want to get to know until where they can reach.

[AntonioLao]

GUILLE,

To be useful, all infinite series must converge. That is to say a limit exist. The limit of all divergent series is infinity.

Series may diverge by marching off to infinity or by oscillating. Divergent series have some curious properties. For example, rearranging the terms of gives both and .

[Guille]

What about non-infinite series. Do these converge or diverge? or another?

[AntonioLao]

What about non-infinite series.

Maybe, they have to do with residue theorem http://mathworld.wolfram.com/ResidueTheorem.html