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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". –  Robert Israel Mar 21 '12 at 22:42
    
@RobertIsrael - But must theorems for convergence is for functions but not sequence that are availble on the internet –  Victor Mar 21 '12 at 22:46
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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. –  André Nicolas Mar 21 '12 at 22:47
    
@AndréNicolas - i think if it is closer or closer uniformly then it will works, right? –  Victor Mar 21 '12 at 22:49
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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. –  André Nicolas Mar 21 '12 at 22:54
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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? –  Victor Mar 21 '12 at 23:08
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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.
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