Assume that the Taylor expansion f(x)=∑∞k=0akxk is convergent for some |x|>1. Then f can be extended in a natural way into the complex domain by writing f(z)=∑∞k=0akzk with z complex and |z|≤1. So we may look at f on the unit circle |z|=1. Consider f as a function of the polar angle ϕ there, i.e., look at the function F(ϕ):=f(eiϕ). This function F is 2π-periodic, and its Fourier expansion is nothing else but F(ϕ)=∑∞k=0akeikϕ where the ak are the Taylor coefficients of the “real” function x↦f(x) we started with.