Let complex numbers \(c_0,c_1,\dots\), \(\sum^n_{v=0} c_v\neq 0\), define a regular Nørlund method; assume throughout a series with partial sums \(s_n(z)= \sum^n_{v=0} a_v z^v\to f(z)\) and radius 1 of convergence. For the Nørlund means \(\sigma_n(z)\) of \((s_n(z))\) and with \(R\geq 1\), let \(M_n[R]\) denote the maximum of \(|\sigma_n(z)|\) for \(|z|=R\). It is proved that \(\limsup M_n[R]^{1/n}= R\) (Theorem 2.1). This equation holds with \(M_n[R]\) being replaced by the maximum of \(|\sigma_n(z)|\) on any (non-singleton) closed arc of the \(R\)-circle (Theorem 2.2). Suppose that \(R>1\); in the case that there exists a subsequence \((\sigma_{n(k)}(z))\) of \((\sigma_n(z))\) such that \(\limsup M_{n(k)}[R]^{1/n(k)}< R\), then, and only then, the series has gaps in the sense of the Hadamard-Ostrowski (Theorem 2.3). Concerning regular Nørlund means with \(c_n> 0\), \(n=0,1,\dots\), F. Leja [Bull. Sci. Math., II. Sér. 54, 239–245 (1930; JFM 56.0910.02)] had proved what covers that, for \(|z|>1\), the series does not converge except for, at most, countably many \(z\) and hence cannot serve for analytic continuation of \(f\). A gap power series may do so compactly beyond the disk of convergence (which is called “overconvergence”). It is shown that if the series under consideration is a gap power series with analytic continuation of \(f\) on a domain exceeding the unique disk, then continuation may be substantiated there by compact convergence of a subsequence \((\sigma_{n(k)}(z))\) (Theorem 3.1). As to \(c_r>0\), a similar result is given for Cesàro means of natural orders (Theorem 3.3).