Taylor Expansions

As in the case of formal power series in \(\mathbb C[[t]]\) , the first step towards the summability of formal power series in \(\mathcal {O}(D_{\rho _1,\ldots ,\rho _n})[[t]]\) is the Poincaré asymptotic which provides, if any exists, the Taylor expansion at \(t=0\) of a holomorphic map \(t\longmapsto u(t,x_0)\) on a given sector \(\Sigma \) , \(x_0\) being an arbitrary fixed point on a convenient polydisc \(D_{r,\ldots ,r}\) .