We introduce the problem of analytical continuation of Taylor series with random coefficients. We identify these series with noisy time series. We build Padé Approximations on these random Taylor series and study systematically the statistical distribution of their poles and zeros in the complex plane. We obtain the unexpected result that these distributions are universal independent of the statistical properties of the input random coefficients. We have tested the universality on Gaussian, Pink, Blank noisy complex input coefficients. We discuss the application of these results to the spectral analysis of noisy time series containing useful signal information and compare it to the traditional Fast Fourier Transform.