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A sequence is called converge if for every next term of the sequence is getting closer to the limit of a number. What is the list of theorem that are able to helping to find out a sequence is converge or not?

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    You really need to look up the definition (not to mention the spelling) of "converge".2012-03-21
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    @RobertIsrael - But must theorems for convergence is for functions but not sequence that are availble on the internet2012-03-21
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    Closer and closer is not quite right. For example the sequence $1,0,1/2,0,1/3,0,1/4,0, \dots$ converges to $0$. But the fifth term ($1/3$) is quite a bit further from $0$ than the fourth term.2012-03-21
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    @AndréNicolas - i think if it is closer or closer uniformly then it will works, right?2012-03-21
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    @Victor: There are other issues, like the terms $1,1/2,1/4,1/8,\dots$ are getting closer and closer to $-17$, but the limit is not $-17$. This is the reason that we give the somewhat convoluted (for every $epsilon$ there exists an $N$ $\dots$) formal definition of convergence. You can find information in Wikipedia and elsewhere on *Convergence Tests*, mainly for series.2012-03-21
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    For just an arbitrary sequence of numbers, there's not much you can say other than the definition and simple corollaries of that. If the sequence is obtained in a particular way (e.g. as partial sums of a series, or integrals of a sequence of functions), there are theorems that may give conditions for the sequence to converge.2012-03-21
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    @AndréNicolas - However i don't know the list of names of these theorems, so may be this would be a trouble for me to knowing the whole thing...2012-03-21
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    @Robert Israel - May the you can give me a list and i could look it up one by one, thanks in advance2012-03-21
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    @Victor: I mentioned the Wikipedia entry on convergence tests. That will give you many names, overwhelmingly for the special kinds of sequences called series. You might also look up the Squeeze Theorem. Another often useful criterion is that a sequence which is increasing and bounded above has a limit, same for decreasing and bounded below.2012-03-21

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Wikipedia has a list of convergence tests for series. You may want to adjust these to looking at first differences to test for convergence of a sequence.

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    Is it too few of them are available?2012-03-21
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Since the theorem is not listed in the link, I'll add it:

Kummer:

Let $b_n$ and $a_n$ be two sequences such that for $n \geq N$, $a_n \wedge b_n >0$.

Then $\sum a_n$ converges if there exists $r$ such that for $n \geq N$ we have that

$$c_n \geq r > 0$$for $c_n = b_n-\dfrac{a_{n+1}}{a_n}b_{n+1}$.

If $c_n < 0$ and $\sum b_n^{-1}$ diverges, so does $\sum a_n$

I find this test fundamental since it is the general case for

  1. D'Alambert's test
  2. Gauss' test
  3. Raabe's test
  4. We know that D'Alambert's criterion is connected to Cauchy's root test.