For a harmonic function on an open subset of real $n$-space, we propose a condition on the Taylor expansion that implies harmonic extension to a larger set, by a result on the size of the domain of convergence of its Taylor series. The result in the $n=2$ case is due to M.\ B\^ocher (1909), and the generalization to $n>2$ is given a mostly elementary proof, using basic facts about multivariable power series.
Adam Coffman. David Legg. Yifei Pan. "A Taylor series condition for harmonic extension.." Real Anal. Exchange 28 (1) 229 - 248, 2002-2003.