Question: This letter "$\varepsilon$" is called epsilon right ? What does it signify in mathematics ?

Answer: 
[Hilbert's epsilon-calculus](http://en.wikipedia.org/wiki/Epsilon_calculus) used the letter $\varepsilon$ to denote a value satisfying a predicate. If $\phi(x)$ is any property, then $\varepsilon x. \phi(x)$ is a term $t$ such that $\phi(t)$ is true, if such $t$ exists. One can define the usual existential and universal quantifiers $\exists$ and $\forall$ in terms of the $\varepsilon$ quantifier:

$$\begin{eqnarray}
\def\hil#1{#1(\varepsilon x. #1(x))}
\exists x.\phi(x) & \equiv & \hil{\phi}\\
\forall x.\phi(x) & \equiv & \phi(\varepsilon x.\lnot\phi(x))
\end{eqnarray}
$$
Here's a not too well-known instance of the use of $\varepsilon$ in mathematics:

One somewhat well-known transformation for accelerating the convergence of a sequence is the [Shanks transformation](http://en.wikipedia.org/wiki/Shanks_transformation) (after [Daniel Shanks](http://en.wikipedia.org/wiki/Daniel_Shanks), who is probably more well-known for his number-theoretic contributions). What the Shanks transformation essentially does, assuming that the sequence given is a sequence of Taylor polynomials evaluated at a certain argument, is to transform this sequence of Taylor approximants into a sequence of [Padé rational approximants](http://en.wikipedia.org/wiki/Pad%C3%A9_approximant).

The Shanks transformation of a sequence can be expressed as a ratio of two determinants, but there is a more efficient realization of this, the [Wynn $\varepsilon$ algorithm](http://dx.doi.org/10.1090/S0025-5718-1956-0084056-6):

$$\varepsilon\_{k+1}^{(n)}=\varepsilon\_{k-1}^{(n+1)}+\frac1{\varepsilon\_{k}^{(n+1)}-\varepsilon\_k^{(n)}}$$

where $\varepsilon\_0^{(n)}=S\_n$ is the sequence to be transformed.