Let \( f(z)=\sum_{k=0}^\infty f_kz^k\) be a power series with radius of convergence \(R_0>0\). Let \(\pi_n=\pi_{n,m}\) be the \(m\)th row of the Padé table of the series (1). The authors prove the extensions of classical theorems of A. Ostrowski and Hadamard: Let (1) be such that its \(m\)th row has Ostrowski type gaps. Then, (1) defines a meromorphic function with a simply connected domain of existence \(G\) in which it has no more than \(m\) poles. Moreover, \(\{\pi_{n_k}\}\) converges to \(f\) \(\sigma\)-almost uniformly inside \(G\).