The author proves two generalizations of \textit{M. Eiermann's} [J. Comput. Appl. Math. 10, 219-227 (1984; Zbl 0538.65011)] sufficient condition for linear summability of power series, one where the summation method is applied to partial sums of the multidimensional power series and one where different summation matrices are used in different variables. The author's theorems only hold in polydiscs in contrast with Eiermann's. He then applies his theorems to convergence of so called Padé type approximants (rational functions that interpolate like polynomials).