Extending valuations to the field of rational functions using pseudo-monotone sequences

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing similar result obtained by Ostrowski in case one-dimensional domains. This accomplished realizing such means pseudo-monotone sequences, generalization pseudo-convergent sequences introduced Chabert. also that rings associated and pseudo-divergent (two classes sequences) roughly correspond, respectively, closed open balls $K$ topology induced $V$.