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when i refer to a convergent sequence i mind also sequence that goes to +- infinity ( but only one of them) that is, given any A , there is an N such that is n>N then $ a_n > A $ etc.

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What is the question? Please make the body of the post self-contained (the subject line is an indication of what the post is about, not part of the post) – Arturo Magidin Aug 7 '11 at 23:20
We can divide the set of sequence in two sets, the set of convergent sequences, and divergent sequence, in the convergent sequence I´ll consider all the "typical convergent sequences" and the sequence that approach to +- infinity, that is, given any A, exist an N such that n>N implies that $ a_n >A $ ( or less than A if the sequence goes to -infinity) . I am asking if exist a good test, that can detect if given a sequence can detect if it is convergent ( in any of the two sense of convergence) I have problems with an exercise that says Let $ a_n $ a sequence such that exist C>0 such that $$ \f – Daniel Aug 7 '11 at 23:42
trying to prove it , im thinking in using Ceraso Stolz , theorem, and proving that afirmation to the limit of $$ a_{n + 1} - a_n $$ someone has an idea with that? – Daniel Aug 8 '11 at 3:28
Please stop adding your comments as answers. You can edit the question and add these comments there, which is what you should be doing. This is not an answer. – Arturo Magidin Aug 8 '11 at 3:41
@Daniel, please try to follow the local customs of the site. – Mariano Suárez-Alvarez♦ Aug 8 '11 at 4:05

closed as not a real question by Qiaochu Yuan Aug 8 '11 at 13:35

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There are many tests for deciding whether a given sequence converges. Which test(s) to use for a particular sequence will depend on what you know about that particular sequence. There is no one-size-fits-all test, except insofar as the definition of convergence could be said to be a test in itself.

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