A series :S = \sum_{n=0}^{\infty} a_n can be written as f(1), where the
function f is defined as :f(z) = \sum_{n=0}^{\infty} a_n z^n. The function f(z) can have
singularities in the
complex plane (
branch point singularities,
poles or
essential singularities), which limit the
radius of convergence of the series. If the point z = 1 is close to or on the boundary of the disk of convergence, the series for S will converge very slowly. One can then improve the convergence of the series by means of a
conformal mapping that moves the singularities such that the point that is mapped to z = 1 ends up deeper in the new disk of convergence. The conformal transform z = \Phi(w) needs to be chosen such that \Phi(0) = 0, and one usually chooses a function that has a finite
derivative at
w = 0. One can assume that \Phi(1) = 1 without loss of generality, as one can always rescale
w to redefine \Phi. We then consider the function :g(w) = f(\Phi(w)). Since \Phi(1) = 1, we have f(1) = g(1). We can obtain the series expansion of g(w) by putting z = \Phi(w) in the series expansion of f(z) because \Phi(0)=0; the first n terms of the series expansion for f(z) will yield the first n terms of the series expansion for g(w) if \Phi'(0) \neq 0. Putting w = 1 in that series expansion will thus yield a series such that if it converges, it will converge to the same value as the original series. ==Non-linear sequence transformations==