A Šleszyňski–Pringsheim theorem for vector valued continued fractions and its optimal error bounds

The first purpose of this work is to prove a convergence theorem for vector valued continued fractions which is an exact extension of a well-known theorem of Šleszyňski–Pringsheim, and our proof is based on the truncation error analysis for such continued fractions. The second purpose is to give refined error bounds for such continued fractions. These bounds are also improvements of the scalar results and the numerical example given explains the effect of our results.