Let U denote the open unit disc
in the complex plane C. A power series of a complex variable with center at the
origin and radius of convergence equal to one will be called a powe7⋅series in U. It is
well-known that the behaviors of a power series outside its circle of convergence are
quite irregular. In particular, it is proved that for each complex number z∗,|z∗| > 1,
and for every closed set E in C, there is a power series in U whose limit set at z∗
is E. The power series Pα(z) =∑(1 − einα)zn,α real, are studied in this
paper. Their peculiar properties seem to suggest that it might be fruitful to
study the irregularities of power series outside their circles of convergence by
probability theory. To study Pα(z), results in diophantine approximation are
obtained.
