What is the connection between Ideal convergerence and 'sequence convergerence'
We say that a sequence of reals is $I$-convergerent to x if and only if for each $\epsilon>0$:
$\left\{n \in \omega \left|x_{n}-x \right| \ge \epsilon \right\} \in I $
classical convergerence: Convergent sequence
It seems that this notion was first introduced in the context of statistics/fuzzy logic; in particular, if you define the ideal to be $$I = \{ A \subset \mathbb{N} : \lim_{n\to\infty} \frac{|A \cap \{1,\dots,n\}|}{n} = 0 \},$$ then you get a notion of "statistical convergence", which is basically a notion of convergence that lets you ignore a bunch of terms in the sequence, and see what the "majority" of elements are doing. More specifically, a sequence $\{a_n\}$ statistically converges to $a$ if the "asymptotic density" of the elements that aren't converging to $t$ is $0$ no matter how far you zoom in, i.e. $$\lim_{n\to\infty} \frac{1}{n}|\{n \in \mathbb{N} : |a_n-a| \geq \epsilon \}| = 0 \text{ for all } \epsilon > 0.$$
On the other hand, if you define $I$ to be all finite subsets of $\mathbb{N}$, you can check that this is the usual notion of sequence convergence.
Sources:
http://thales.doa.fmph.uniba.sk/katc/publ/salat_i_cong.pdf http://arxiv.org/pdf/math/0612179.pdf