Peter Borwein, Michael Coons, Transcendence of Power Series for Some Number Theoretic Functions, Proceedings of the American Mathematical Society, Vol. 137, No. 4 (Apr., 2009), pp. 1303-1305 ...

A POWER series of the form?an(z - z0)n which converges at more than one point, converges inside a circle centre z0 and coincides with the series obtained by applying Taylor's theorem to the sum ...