]> $${\psi }^{\left(n\right)}\left(x\right)=\frac{{\left(-1\right)}^{n-1}n!}{x^{n+1}}+\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^{k+n+1}\left(k+n\right)!\zeta \left(k+n+1\right)x^k}{k!}$$