The Vivanti–Pringsheim theorem is a mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was originally formulated by Giulio Vivanti in 1893 and proved in the following year by Alfred Pringsheim.
More precisely the theorem states the following:
A complex function defined by a power series 
formula_1 
with non-negative real coefficients formula_2 and a radius of convergence formula_3 has a singularity at formula_4.
A simple example is the (complex) geometric series 
formula_5
with a singularity at formula_6.