  This paper demonstrates some connections between the coefficients of a Taylor
series $f(z)=\ds\sum_{n=0}^\infty a_n z^n$ and singularities of the function.
There are many known results of this type, for example, counting the number of
poles on the circle of convergence, and doing convergence or overconvergence
for $f$ on any arc of holomorphy. A new approach proposed here is that these
kinds of results are extended by relaxing the classical conditions for
singularities and convergence theorems. This is done by allowing the
coefficients to be sufficiently small instead of being zero.
