We show that any formal embedding sending a real-analytic strongly pseudoconvex hypersurface M ⊂ ℂN, N > 1, into another such hypersurface M′ ⊂ ℂN+1 is convergent.
"Convergence of Formal Embeddings Between Real-Analytic Hypersurfaces in Codimension One." J. Differential Geom. 62 (1) 163 - 173, September, 2002. https://doi.org/10.4310/jdg/1090425533