General question about convergence of series

AI Thread Summary
A sequence converges if it has a finite limit, while it diverges if it does not. For a series, if it converges, the limit of its terms approaches zero as n approaches infinity. However, a limit of zero does not guarantee convergence; it only indicates a possibility. If the limit of the terms does not equal zero, the series diverges. The definitions of convergence and divergence are fundamentally tied to the existence of limits.
RadiationX
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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.
 
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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.
 
Are you talking about sequences or series, RadiationX?

By definition, if a sequence has a finite limit, then it converges to that limit.
 
yes i was talking about sequences
 
Yes. "Converge" is DEFINED as "has a limit". "Diverge" is defined as "does not have a limit".
 

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