The present study focuses on the determination of an asymptotic Graetz series for the demanding upstream sub-region of the Graetz problem where the Graetz series converges slowly. The Method Of Lines (MOL) is the computational procedure that transforms the descriptive two–dimensional energy equation in the axial and radial variables into an allied system of ordinary differential equations of first order in the axial variable. An allied system of ordinary differential equations of first order having three equations is deduced and solved analytically with the robust eigenvalue method of linear algebra theory. The strategy to follow revolves around the level of difficulty in the determination of the eigenquantities with MOL as compared to the level of difficulty confronted by Nusselt and Jakob in the past. From a qualitative perspective, the obtained three-term MOL series representative of the asymptotic mean bulk temperature in the sub-region near the origin exhibits good convergence patterns. The three-term MOL series supplements the truncated three-term Graetz series depictive of the asymptotic mean bulk temperature in the downstream sub-region far from the origin.