What is the list of theorem that are able to find out a sequence is converge or not?

Question

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?

Answer 1

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.

Answer 2

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.