Do sequences with limits always converge?

[RadiationX]

In general, is it true that if a sequence has a limit that it converges and if it does not have a limit that it diverges?when i say have a limit i mean that the limit exists.

 

[Jameson]

Given $$\sum_{n=0}^{\infty} a_n$$

if the series converges, then

$$\lim_{n\rightarrow\infty}a_n = 0$$

This does not mean that if the limit = 0 it converges, but that it has a possibility to converge. If the limit does not equal 0, the series diverges.

 

[James R]

Are you talking about sequences or series, RadiationX?

By definition, if a sequence has a finite limit, then it converges to that limit.

 

[JimmI_Hendrix]

yes i was talking about sequences

 

[HallsofIvy]

Yes. "Converge" is DEFINED as "has a limit". "Diverge" is defined as "does not have a limit".